Unfortunately, the accepted answer as well the other answer are wrong.
You have done two mistakes in parsing:
- You have swapped 
MM with dd. 
- You have used 
H instead of h to parse a date-time string with AM/PM marker. The symbol, H is used to parse a time string in 24-Hour format. Another important point to consider is that PM is written as p.m. in Locale.CANADA. For your time string, you can use Locale.US which specifies it as PM. 
Thus, you can use a parser like the one given below:
new SimpleDateFormat("M/d/u h:m:s a", Locale.US)
where a has been used to parse AM/PM.
Using this parser will fix your problem but the legacy date-time API (java.util date-time types and their formatting API, SimpleDateFormat) is outdated and error-prone. It is recommended to stop using it completely and switch to java.time, the modern date-time API*.
Demo using modern date-time API:
import java.time.LocalDateTime;
import java.time.format.DateTimeFormatter;
import java.util.Locale;
public class Main {
    public static void main(String args[]) {
        String convertDate = "03/19/2014 5:30:10 PM";
        DateTimeFormatter dtfInput = DateTimeFormatter.ofPattern("M/d/u h:m:s a", Locale.US);
        LocalDateTime ldt = LocalDateTime.parse(convertDate, dtfInput);
        System.out.printf("Year: %d, Month: %d, Day: %d%n", ldt.getYear(), ldt.getMonthValue(), ldt.getDayOfMonth());
        // Or, using DateTimeFormatter
        DateTimeFormatter dtfOutput = DateTimeFormatter.ofPattern("'Year:' uuuu, 'Month:' M, 'Day:' d", Locale.US);
        System.out.println(ldt.format(dtfOutput));
    }
}
Output:
Year: 2014, Month: 3, Day: 19
Year: 2014, Month: 3, Day: 19
Learn more about the the modern date-time API* from Trail: Date Time.
* For any reason, if you have to stick to Java 6 or Java 7, you can use ThreeTen-Backport which backports most of the java.time functionality to Java 6 & 7. If you are working for an Android project and your Android API level is still not compliant with Java-8, check Java 8+ APIs available through desugaring and How to use ThreeTenABP in Android Project.