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?
Use:
https://www.googleapis.com/youtube/v3/videoCategories?part=snippet,id&maxResults=100®ionCode=us&key=**Your YouTube ID**
Above is the link. Using that, you can find the YouTube characteristics of videos. After finding characteristics, you can get videos of the selected category. After then you can find selected video images using Asaph's answer.
Try the above approach and you can parse everything from the YouTube API.