Event Extension
Event extension is very appropriate. It is a port of Libevent library which is designed for event-driven I/O, mainly for networking.
I have written a sample HTTP client that allows to schedule a number of 
HTTP requests and run them asynchronously.
This is a sample HTTP client class based on Event extension.
The class allows to schedule a number of HTTP requests, then run them asynchronously. 
http-client.php
<?php
class MyHttpClient {
  /// @var EventBase
  protected $base;
  /// @var array Instances of EventHttpConnection
  protected $connections = [];
  public function __construct() {
    $this->base = new EventBase();
  }
  /**
   * Dispatches all pending requests (events)
   *
   * @return void
   */
  public function run() {
    $this->base->dispatch();
  }
  public function __destruct() {
    // Destroy connection objects explicitly, don't wait for GC.
    // Otherwise, EventBase may be free'd earlier.
    $this->connections = null;
  }
  /**
   * @brief Adds a pending HTTP request
   *
   * @param string $address Hostname, or IP
   * @param int $port Port number
   * @param array $headers Extra HTTP headers
   * @param int $cmd A EventHttpRequest::CMD_* constant
   * @param string $resource HTTP request resource, e.g. '/page?a=b&c=d'
   *
   * @return EventHttpRequest|false
   */
  public function addRequest($address, $port, array $headers,
    $cmd = EventHttpRequest::CMD_GET, $resource = '/')
  {
    $conn = new EventHttpConnection($this->base, null, $address, $port);
    $conn->setTimeout(5);
    $req = new EventHttpRequest([$this, '_requestHandler'], $this->base);
    foreach ($headers as $k => $v) {
      $req->addHeader($k, $v, EventHttpRequest::OUTPUT_HEADER);
    }
    $req->addHeader('Host', $address, EventHttpRequest::OUTPUT_HEADER);
    $req->addHeader('Connection', 'close', EventHttpRequest::OUTPUT_HEADER);
    if ($conn->makeRequest($req, $cmd, $resource)) {
      $this->connections []= $conn;
      return $req;
    }
    return false;
  }
  /**
   * @brief Handles an HTTP request
   *
   * @param EventHttpRequest $req
   * @param mixed $unused
   *
   * @return void
   */
  public function _requestHandler($req, $unused) {
    if (is_null($req)) {
      echo "Timed out\n";
    } else {
      $response_code = $req->getResponseCode();
      if ($response_code == 0) {
        echo "Connection refused\n";
      } elseif ($response_code != 200) {
        echo "Unexpected response: $response_code\n";
      } else {
        echo "Success: $response_code\n";
        $buf = $req->getInputBuffer();
        echo "Body:\n";
        while ($s = $buf->readLine(EventBuffer::EOL_ANY)) {
          echo $s, PHP_EOL;
        }
      }
    }
  }
}
$address = "my-host.local";
$port = 80;
$headers = [ 'User-Agent' => 'My-User-Agent/1.0', ];
$client = new MyHttpClient();
// Add pending requests
for ($i = 0; $i < 10; $i++) {
  $client->addRequest($address, $port, $headers,
    EventHttpRequest::CMD_GET, '/test.php?a=' . $i);
}
// Dispatch pending requests
$client->run();
test.php
This is a sample script on the server side. 
<?php
echo 'GET: ', var_export($_GET, true), PHP_EOL;
echo 'User-Agent: ', $_SERVER['HTTP_USER_AGENT'] ?? '(none)', PHP_EOL;
Usage
php http-client.php
Sample Output
Success: 200
Body:
GET: array (
  'a' => '1',
)
User-Agent: My-User-Agent/1.0
Success: 200
Body:
GET: array (
  'a' => '0',
)
User-Agent: My-User-Agent/1.0
Success: 200
Body:
GET: array (
  'a' => '3',
)
...
(Trimmed.)
Note, the code is designed for long-term processing in the CLI SAPI.
For custom protocols, consider using low-level API, i.e. buffer events, buffers. For SSL/TLS communications, I would recommend the low-level API in conjunction with Event's ssl context. Examples:
Although Libevent's HTTP API is simple, it is not as flexible as buffer events. For example, the HTTP API currently doesn't support custom HTTP methods. But it is possible to implement virtually any protocol using the low-level API.
Ev Extension
I have also written a sample of another HTTP client using Ev extension with sockets in non-blocking mode. The code is slightly more verbose than the sample based on Event, because Ev is a general purpose event loop. It doesn't provide network-specific functions, but its EvIo watcher is capable of listening to a file descriptor encapsulated into the socket resource, in particular.
This is a sample HTTP client based on Ev extension.
Ev extension implements a simple yet powerful general purpose event loop. It doesn't provide network-specific watchers, but its I/O watcher can be used for asynchronous processing of sockets.
The following code shows how HTTP requests can be scheduled for parallel processing.
http-client.php
<?php
class MyHttpRequest {
  /// @var MyHttpClient
  private $http_client;
  /// @var string
  private $address;
  /// @var string HTTP resource such as /page?get=param
  private $resource;
  /// @var string HTTP method such as GET, POST etc.
  private $method;
  /// @var int
  private $service_port;
  /// @var resource Socket
  private $socket;
  /// @var double Connection timeout in seconds.
  private $timeout = 10.;
  /// @var int Chunk size in bytes for socket_recv()
  private $chunk_size = 20;
  /// @var EvTimer
  private $timeout_watcher;
  /// @var EvIo
  private $write_watcher;
  /// @var EvIo
  private $read_watcher;
  /// @var EvTimer
  private $conn_watcher;
  /// @var string buffer for incoming data
  private $buffer;
  /// @var array errors reported by sockets extension in non-blocking mode.
  private static $e_nonblocking = [
    11, // EAGAIN or EWOULDBLOCK
    115, // EINPROGRESS
  ];
  /**
   * @param MyHttpClient $client
   * @param string $host Hostname, e.g. google.co.uk
   * @param string $resource HTTP resource, e.g. /page?a=b&c=d
   * @param string $method HTTP method: GET, HEAD, POST, PUT etc.
   * @throws RuntimeException
   */
  public function __construct(MyHttpClient $client, $host, $resource, $method) {
    $this->http_client = $client;
    $this->host        = $host;
    $this->resource    = $resource;
    $this->method      = $method;
    // Get the port for the WWW service
    $this->service_port = getservbyname('www', 'tcp');
    // Get the IP address for the target host
    $this->address = gethostbyname($this->host);
    // Create a TCP/IP socket
    $this->socket = socket_create(AF_INET, SOCK_STREAM, SOL_TCP);
    if (!$this->socket) {
      throw new RuntimeException("socket_create() failed: reason: " .
        socket_strerror(socket_last_error()));
    }
    // Set O_NONBLOCK flag
    socket_set_nonblock($this->socket);
    $this->conn_watcher = $this->http_client->getLoop()
      ->timer(0, 0., [$this, 'connect']);
  }
  public function __destruct() {
    $this->close();
  }
  private function freeWatcher(&$w) {
    if ($w) {
      $w->stop();
      $w = null;
    }
  }
  /**
   * Deallocates all resources of the request
   */
  private function close() {
    if ($this->socket) {
      socket_close($this->socket);
      $this->socket = null;
    }
    $this->freeWatcher($this->timeout_watcher);
    $this->freeWatcher($this->read_watcher);
    $this->freeWatcher($this->write_watcher);
    $this->freeWatcher($this->conn_watcher);
  }
  /**
   * Initializes a connection on socket
   * @return bool
   */
  public function connect() {
    $loop = $this->http_client->getLoop();
    $this->timeout_watcher = $loop->timer($this->timeout, 0., [$this, '_onTimeout']);
    $this->write_watcher = $loop->io($this->socket, Ev::WRITE, [$this, '_onWritable']);
    return socket_connect($this->socket, $this->address, $this->service_port);
  }
  /**
   * Callback for timeout (EvTimer) watcher
   */
  public function _onTimeout(EvTimer $w) {
    $w->stop();
    $this->close();
  }
  /**
   * Callback which is called when the socket becomes wriable
   */
  public function _onWritable(EvIo $w) {
    $this->timeout_watcher->stop();
    $w->stop();
    $in = implode("\r\n", [
      "{$this->method} {$this->resource} HTTP/1.1",
      "Host: {$this->host}",
      'Connection: Close',
    ]) . "\r\n\r\n";
    if (!socket_write($this->socket, $in, strlen($in))) {
      trigger_error("Failed writing $in to socket", E_USER_ERROR);
      return;
    }
    $loop = $this->http_client->getLoop();
    $this->read_watcher = $loop->io($this->socket,
      Ev::READ, [$this, '_onReadable']);
    // Continue running the loop
    $loop->run();
  }
  /**
   * Callback which is called when the socket becomes readable
   */
  public function _onReadable(EvIo $w) {
    // recv() 20 bytes in non-blocking mode
    $ret = socket_recv($this->socket, $out, 20, MSG_DONTWAIT);
    if ($ret) {
      // Still have data to read. Append the read chunk to the buffer.
      $this->buffer .= $out;
    } elseif ($ret === 0) {
      // All is read
      printf("\n<<<<\n%s\n>>>>", rtrim($this->buffer));
      fflush(STDOUT);
      $w->stop();
      $this->close();
      return;
    }
    // Caught EINPROGRESS, EAGAIN, or EWOULDBLOCK
    if (in_array(socket_last_error(), static::$e_nonblocking)) {
      return;
    }
    $w->stop();
    $this->close();
  }
}
/////////////////////////////////////
class MyHttpClient {
  /// @var array Instances of MyHttpRequest
  private $requests = [];
  /// @var EvLoop
  private $loop;
  public function __construct() {
    // Each HTTP client runs its own event loop
    $this->loop = new EvLoop();
  }
  public function __destruct() {
    $this->loop->stop();
  }
  /**
   * @return EvLoop
   */
  public function getLoop() {
    return $this->loop;
  }
  /**
   * Adds a pending request
   */
  public function addRequest(MyHttpRequest $r) {
    $this->requests []= $r;
  }
  /**
   * Dispatches all pending requests
   */
  public function run() {
    $this->loop->run();
  }
}
/////////////////////////////////////
// Usage
$client = new MyHttpClient();
foreach (range(1, 10) as $i) {
  $client->addRequest(new MyHttpRequest($client, 'my-host.local', '/test.php?a=' . $i, 'GET'));
}
$client->run();
Testing
Suppose http://my-host.local/test.php script is printing the dump of $_GET:
<?php
echo 'GET: ', var_export($_GET, true), PHP_EOL;
Then the output of php http-client.php command will be similar to the following:
<<<<
HTTP/1.1 200 OK
Server: nginx/1.10.1
Date: Fri, 02 Dec 2016 12:39:54 GMT
Content-Type: text/html; charset=UTF-8
Transfer-Encoding: chunked
Connection: close
X-Powered-By: PHP/7.0.13-pl0-gentoo
1d
GET: array (
  'a' => '3',
)
0
>>>>
<<<<
HTTP/1.1 200 OK
Server: nginx/1.10.1
Date: Fri, 02 Dec 2016 12:39:54 GMT
Content-Type: text/html; charset=UTF-8
Transfer-Encoding: chunked
Connection: close
X-Powered-By: PHP/7.0.13-pl0-gentoo
1d
GET: array (
  'a' => '2',
)
0
>>>>
...
(trimmed)
Note, in PHP 5 the sockets extension may log warnings for EINPROGRESS, EAGAIN, and EWOULDBLOCK errno values. It is possible to turn off the logs with
error_reporting(E_ERROR);
Concerning "the Rest" of the Code
I just want to do something like file_get_contents(), but not wait for the request to finish before executing the rest of my code.
The code that is supposed to run in parallel with the network requests can be executed within a the callback of an Event timer, or Ev's idle watcher, for instance. You can easily figure it out by watching the samples mentioned above. Otherwise, I'll add another example :)