This seems to work just fine:

jQuery(document).ready(function ($) {
    $('.video1').magnificPopup({
       items: {
          src: '<iframe width="560" height="315" src="https://www.youtube.com/embed/_ziUhNerFMI?autoplay=1&rel=0" frameborder="0" allowfullscreen></iframe>',
          type: 'inline'
      }
    });
});

.video1 {
  background-image: url('http://placehold.it/180x180');
  background-repeat: no-repeat;
  background-size: contain;
  height: 180px;
  width: 180px;
  display: block;
  outline: none;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>
<script src="http://cdnjs.cloudflare.com/ajax/libs/magnific-popup.js/1.0.0/jquery.magnific-popup.min.js"></script>
<a class="video1" href="https://www.youtube.com/watch?v=_ziUhNerFMI"></a>

By default Magnific Popup supports only one type of URL for each service so I extent it for support almost every video URL type of YouTube/Vimeo:

http://dimsemenov.com/plugins/magnific-popup/documentation.html#iframe-type

$('.my-selector').magnificPopup({
    type: 'iframe',
    iframe: {
        patterns: {
            youtube: {
                index: 'youtube.com/', 
                id: function(url) {        
                    var m = url.match(/[\\?\\&]v=([^\\?\\&]+)/);
                    if ( !m || !m[1] ) return null;
                    return m[1];
                },
                src: '//www.youtube.com/embed/%id%?autoplay=1'
            },
            vimeo: {
                index: 'vimeo.com/', 
                id: function(url) {        
                    var m = url.match(/(https?:\/\/)?(www.)?(player.)?vimeo.com\/([a-z]*\/)*([0-9]{6,11})[?]?.*/);
                    if ( !m || !m[5] ) return null;
                    return m[5];
                },
                src: '//player.vimeo.com/video/%id%?autoplay=1'
            }
        }
    }
});

Just copy those two property (iframe, type) and add them to your Magnific Popup.