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?
In YouTube Data API v3, you can get video's thumbnails with the videos->list function. From snippet.thumbnails.(key), you can pick the default, medium or high resolution thumbnail, and get its width, height and URL.
You can also update thumbnails with the thumbnails->set functionality.
For examples, you can check out the YouTube API Samples project. (PHP ones.)