Each YouTube video has four generated images. They are predictably formatted as follows:
https://img.youtube.com/vi/<insert-youtube-video-id-here>/0.jpg
https://img.youtube.com/vi/<insert-youtube-video-id-here>/1.jpg
https://img.youtube.com/vi/<insert-youtube-video-id-here>/2.jpg
https://img.youtube.com/vi/<insert-youtube-video-id-here>/3.jpg
The first one in the list is a full size image and others are thumbnail images. The default thumbnail image (i.e., one of 1.jpg, 2.jpg, 3.jpg) is:
https://img.youtube.com/vi/<insert-youtube-video-id-here>/default.jpg
For the high quality version of the thumbnail use a URL similar to this:
https://img.youtube.com/vi/<insert-youtube-video-id-here>/hqdefault.jpg
There is also a medium quality version of the thumbnail, using a URL similar to the HQ:
https://img.youtube.com/vi/<insert-youtube-video-id-here>/mqdefault.jpg
For the standard definition version of the thumbnail, use a URL similar to this:
https://img.youtube.com/vi/<insert-youtube-video-id-here>/sddefault.jpg
For the maximum resolution version of the thumbnail use a URL similar to this:
https://img.youtube.com/vi/<insert-youtube-video-id-here>/maxresdefault.jpg
All of the above URLs are available over HTTP too. Additionally, the slightly shorter hostname i3.ytimg.com works in place of img.youtube.com in the example URLs above.
Alternatively, you can use the YouTube Data API (v3) to get thumbnail images.
Best Answer
I just found out that the following works:
Note: the time must be an integer number of seconds (e.g. 119, not 1m59s).