You can embed a part of YouTube video using the YouTube Player API. The API offers loadVideoById and cueVideoById function both of which allow you to specify a start and end time for the embedded video.

<!-- We specify the various video parameters in a DIV tag -->

<div videoID="t2MOwQ089eQ" startTime="1330" endTime="1466" height="309" width="550" id="youtube-player"></div>

<script>
  /*

  YouTube Embed Code

  Author:   Amit Agarwal
  Blog:     www.labnol.org
  Date:     04/15/2013

  */

  var node = 'youtube-player';

  // Append the YouTube IFRAME API Script
  var youtube = document.createElement('script');
  youtube.type = 'text/javascript';
  youtube.src = '//www.youtube.com/iframe_api';

  var s = document.getElementsByTagName('script')[0];
  s.parentNode.insertBefore(youtube, s);

  var player;

  // Read all the parameter of the DIV tag
  var params = document.getElementById(node);

  var startTime = params.getAttribute('startTime');
  var endTime = params.getAttribute('endTime');
  var videoID = params.getAttribute('videoID');
  var playerHeight = params.getAttribute('height');
  var playerWidth = params.getAttribute('width');

  // Prepare the YouTube Player
  // We set rel=0 and showinfo=1 to hide related videos & info bar

  function onYouTubeIframeAPIReady() {
    player = new YT.Player(node, {
      height: playerHeight,
      width: playerWidth,
      playerVars: { rel: 0, showinfo: 0, hidecontrols: 1 },
      events: {
        onReady: loadVideo,
      },
    });
  }

  // When the player is ready, we load the video
  // Using cueVideoById and not loadVideoById as the
  // former function will not autoplay the video.
  function loadVideo(e) {
    e.target.cueVideoById({
      videoId: videoID,
      startSeconds: startTime,
      endSeconds: endTime,
    });
  }
</script>