If I have a YouTube video URL, is there any way to use PHP and cURL to get the associated thumbnail from the YouTube API?
If all you want to do is search YouTube and get associated properties:
Get a public API -- This link gives a good direction
Use below query string. The search query (denoted by q=) in the URL string is stackoverflow for example purposes. YouTube will then send you back a JSON reply where you can then parse for Thumbnail, Snippet, Author, etc.
https://www.googleapis.com/youtube/v3/search?part=id%2Csnippet&maxResults=50&q=stackoverflow&key=YOUR_API_KEY_HERE