Please note that if the web service supports caching output, you don't have to use CacheRequest below, because Volley will automatically cache.
For your issue, I use some codes inside parseCacheHeaders (and refering to @oleksandr_yefremov's codes). The following code I have tested. Of course, can use for JsonArrayRequest also. Hope this help!
    JsonObjectRequest jsonObjectRequest = new JsonObjectRequest(0, mUrl, new Response.Listener<JSONObject>() {
        @Override
        public void onResponse(JSONObject response) {
            try {
                mTextView.setText(response.toString(5));
            } catch (JSONException e) {
                mTextView.setText(e.toString());
            }
        }
    }, new Response.ErrorListener() {
        @Override
        public void onErrorResponse(VolleyError error) {
        }
    }) {
        @Override
        protected Response<JSONObject> parseNetworkResponse(NetworkResponse response) {
            try {                    
                Cache.Entry cacheEntry = HttpHeaderParser.parseCacheHeaders(response);
                if (cacheEntry == null) {
                    cacheEntry = new Cache.Entry();
                }
                final long cacheHitButRefreshed = 3 * 60 * 1000; // in 3 minutes cache will be hit, but also refreshed on background
                final long cacheExpired = 24 * 60 * 60 * 1000; // in 24 hours this cache entry expires completely
                long now = System.currentTimeMillis();
                final long softExpire = now + cacheHitButRefreshed;
                final long ttl = now + cacheExpired;
                cacheEntry.data = response.data;
                cacheEntry.softTtl = softExpire;
                cacheEntry.ttl = ttl;
                String headerValue;
                headerValue = response.headers.get("Date");
                if (headerValue != null) {
                    cacheEntry.serverDate = HttpHeaderParser.parseDateAsEpoch(headerValue);
                }
                headerValue = response.headers.get("Last-Modified");
                if (headerValue != null) {
                    cacheEntry.lastModified = HttpHeaderParser.parseDateAsEpoch(headerValue);
                }
                cacheEntry.responseHeaders = response.headers;
                final String jsonString = new String(response.data,
                        HttpHeaderParser.parseCharset(response.headers));
                return Response.success(new JSONObject(jsonString), cacheEntry);
            } catch (UnsupportedEncodingException e) {
                return Response.error(new ParseError(e));
            } catch (JSONException e) {
                return Response.error(new ParseError(e));
            }
        }
        @Override
        protected void deliverResponse(JSONObject response) {
            super.deliverResponse(response);
        }
        @Override
        public void deliverError(VolleyError error) {
            super.deliverError(error);
        }
        @Override
        protected VolleyError parseNetworkError(VolleyError volleyError) {
            return super.parseNetworkError(volleyError);
        }
    };
    MySingleton.getInstance(this).addToRequestQueue(jsonObjectRequest);
UPDATE:
If you want a base class, refer to the following codes:
public class CacheRequest extends Request<NetworkResponse> {
    private final Response.Listener<NetworkResponse> mListener;
    private final Response.ErrorListener mErrorListener;
    public CacheRequest(int method, String url, Response.Listener<NetworkResponse> listener, Response.ErrorListener errorListener) {
        super(method, url, errorListener);
        this.mListener = listener;
        this.mErrorListener = errorListener;
    }
    @Override
    protected Response<NetworkResponse> parseNetworkResponse(NetworkResponse response) {
        Cache.Entry cacheEntry = HttpHeaderParser.parseCacheHeaders(response);
        if (cacheEntry == null) {
            cacheEntry = new Cache.Entry();
        }
        final long cacheHitButRefreshed = 3 * 60 * 1000; // in 3 minutes cache will be hit, but also refreshed on background
        final long cacheExpired = 24 * 60 * 60 * 1000; // in 24 hours this cache entry expires completely
        long now = System.currentTimeMillis();
        final long softExpire = now + cacheHitButRefreshed;
        final long ttl = now + cacheExpired;
        cacheEntry.data = response.data;
        cacheEntry.softTtl = softExpire;
        cacheEntry.ttl = ttl;
        String headerValue;
        headerValue = response.headers.get("Date");
        if (headerValue != null) {
            cacheEntry.serverDate = HttpHeaderParser.parseDateAsEpoch(headerValue);
        }
        headerValue = response.headers.get("Last-Modified");
        if (headerValue != null) {
            cacheEntry.lastModified = HttpHeaderParser.parseDateAsEpoch(headerValue);
        }
        cacheEntry.responseHeaders = response.headers;
        return Response.success(response, cacheEntry);
    }
    @Override
    protected void deliverResponse(NetworkResponse response) {
        mListener.onResponse(response);
    }
    @Override
    protected VolleyError parseNetworkError(VolleyError volleyError) {
        return super.parseNetworkError(volleyError);
    }
    @Override
    public void deliverError(VolleyError error) {
        mErrorListener.onErrorResponse(error);
    }
}
Then in MainActivity, you can call like this
CacheRequest cacheRequest = new CacheRequest(0, mUrl, new Response.Listener<NetworkResponse>() {
        @Override
        public void onResponse(NetworkResponse response) {
            try {
                final String jsonString = new String(response.data,
                        HttpHeaderParser.parseCharset(response.headers));
                JSONObject jsonObject = new JSONObject(jsonString);
                mTextView.setText(jsonObject.toString(5));
            } catch (UnsupportedEncodingException | JSONException e) {
                e.printStackTrace();
            }
        }
    }, new Response.ErrorListener() {
        @Override
        public void onErrorResponse(VolleyError error) {
            mTextView.setText(error.toString());
        }
    });
    MySingleton.getInstance(this).addToRequestQueue(cacheRequest);
UPDATE WITH FULL SOURCE CODE:
MainActivity.java:
package com.example.cachevolley;
import android.content.Context;
import android.os.Bundle;
import android.support.v7.app.AppCompatActivity;
import android.view.Menu;
import android.view.MenuItem;
import android.widget.TextView;
import android.widget.Toast;
import com.android.volley.Cache;
import com.android.volley.NetworkResponse;
import com.android.volley.Request;
import com.android.volley.RequestQueue;
import com.android.volley.Response;
import com.android.volley.VolleyError;
import com.android.volley.toolbox.HttpHeaderParser;
import com.android.volley.toolbox.Volley;
import org.json.JSONException;
import org.json.JSONObject;
import java.io.UnsupportedEncodingException;
public class MainActivity extends AppCompatActivity {
    private final Context mContext = this;
    @Override
    protected void onCreate(Bundle savedInstanceState) {
        super.onCreate(savedInstanceState);
        setContentView(R.layout.activity_main);
        final TextView textView = (TextView) findViewById(R.id.textView);
        RequestQueue queue = Volley.newRequestQueue(this);
        String url = "http://192.168.0.100/apitest";
        CacheRequest cacheRequest = new CacheRequest(0, url, new Response.Listener<NetworkResponse>() {
            @Override
            public void onResponse(NetworkResponse response) {
                try {
                    final String jsonString = new String(response.data,
                            HttpHeaderParser.parseCharset(response.headers));
                    JSONObject jsonObject = new JSONObject(jsonString);
                    textView.setText(jsonObject.toString(5));
                    Toast.makeText(mContext, "onResponse:\n\n" + jsonObject.toString(), Toast.LENGTH_SHORT).show();
                } catch (UnsupportedEncodingException | JSONException e) {
                    e.printStackTrace();
                }
            }
        }, new Response.ErrorListener() {
            @Override
            public void onErrorResponse(VolleyError error) {
                Toast.makeText(mContext, "onErrorResponse:\n\n" + error.toString(), Toast.LENGTH_SHORT).show();
            }
        });
        // Add the request to the RequestQueue.
        queue.add(cacheRequest);
    }
    @Override
    public boolean onCreateOptionsMenu(Menu menu) {
        // Inflate the menu; this adds items to the action bar if it is present.
        getMenuInflater().inflate(R.menu.menu_main, menu);
        return true;
    }
    @Override
    public boolean onOptionsItemSelected(MenuItem item) {
        // Handle action bar item clicks here. The action bar will
        // automatically handle clicks on the Home/Up button, so long
        // as you specify a parent activity in AndroidManifest.xml.
        int id = item.getItemId();
        //noinspection SimplifiableIfStatement
        if (id == R.id.action_settings) {
            return true;
        }
        return super.onOptionsItemSelected(item);
    }
    private class CacheRequest extends Request<NetworkResponse> {
        private final Response.Listener<NetworkResponse> mListener;
        private final Response.ErrorListener mErrorListener;
        public CacheRequest(int method, String url, Response.Listener<NetworkResponse> listener, Response.ErrorListener errorListener) {
            super(method, url, errorListener);
            this.mListener = listener;
            this.mErrorListener = errorListener;
        }
        @Override
        protected Response<NetworkResponse> parseNetworkResponse(NetworkResponse response) {
            Cache.Entry cacheEntry = HttpHeaderParser.parseCacheHeaders(response);
            if (cacheEntry == null) {
                cacheEntry = new Cache.Entry();
            }
            final long cacheHitButRefreshed = 3 * 60 * 1000; // in 3 minutes cache will be hit, but also refreshed on background
            final long cacheExpired = 24 * 60 * 60 * 1000; // in 24 hours this cache entry expires completely
            long now = System.currentTimeMillis();
            final long softExpire = now + cacheHitButRefreshed;
            final long ttl = now + cacheExpired;
            cacheEntry.data = response.data;
            cacheEntry.softTtl = softExpire;
            cacheEntry.ttl = ttl;
            String headerValue;
            headerValue = response.headers.get("Date");
            if (headerValue != null) {
                cacheEntry.serverDate = HttpHeaderParser.parseDateAsEpoch(headerValue);
            }
            headerValue = response.headers.get("Last-Modified");
            if (headerValue != null) {
                cacheEntry.lastModified = HttpHeaderParser.parseDateAsEpoch(headerValue);
            }
            cacheEntry.responseHeaders = response.headers;
            return Response.success(response, cacheEntry);
        }
        @Override
        protected void deliverResponse(NetworkResponse response) {
            mListener.onResponse(response);
        }
        @Override
        protected VolleyError parseNetworkError(VolleyError volleyError) {
            return super.parseNetworkError(volleyError);
        }
        @Override
        public void deliverError(VolleyError error) {
            mErrorListener.onErrorResponse(error);
        }
    }
}
Manifest file:
<?xml version="1.0" encoding="utf-8"?>
<manifest xmlns:android="http://schemas.android.com/apk/res/android"
    package="com.example.cachevolley" >
    <uses-permission android:name="android.permission.INTERNET" />
    <application
        android:allowBackup="true"
        android:icon="@mipmap/ic_launcher"
        android:label="@string/app_name"
        android:theme="@style/AppTheme" >
        <activity
            android:name=".MainActivity"
            android:label="@string/app_name" >
            <intent-filter>
                <action android:name="android.intent.action.MAIN" />
                <category android:name="android.intent.category.LAUNCHER" />
            </intent-filter>
        </activity>
    </application>
</manifest>
Layout file:
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
    xmlns:tools="http://schemas.android.com/tools"
    android:layout_width="match_parent"
    android:layout_height="match_parent"
    android:paddingBottom="@dimen/activity_vertical_margin"
    android:paddingLeft="@dimen/activity_horizontal_margin"
    android:paddingRight="@dimen/activity_horizontal_margin"
    android:paddingTop="@dimen/activity_vertical_margin"
    tools:context=".MainActivity">
    <TextView
        android:id="@+id/textView"
        android:layout_width="wrap_content"
        android:layout_height="wrap_content"
        android:text="@string/hello_world" />
</RelativeLayout>