Just to add/expand on the solutions given, I feel it is necessary to note that, as I had this problem myself, one can actually grab multiple YouTube videos content, in this case, thumbnails, with one HTTP request:
Using a Rest Client, in this case, HTTPFUL, you can do something like this:
<?php
header("Content-type", "application/json");
//download the httpfull.phar file from http://phphttpclient.com
include("httpful.phar");
$youtubeVidIds= array("nL-rk4bgJWU", "__kupr7KQos", "UCSynl4WbLQ", "joPjqEGJGqU", "PBwEBjX3D3Q");
$response = \Httpful\Request::get("https://www.googleapis.com/youtube/v3/videos?key=YourAPIKey4&part=snippet&id=".implode (",",$youtubeVidIds)."")
->send();
print ($response);
?>