I want to embed a video from YouTube that is not mine (so I can not change it at YouTube). The video has a thumbnail that is not representative for the video (I refer to the
No. Most YouTube videos only have one pre-generated "poster" thumbnail (480x360). They usually have several other lower resolution thumbnails (120x90). So even if there were an embedding parameter to use an alternate poster image (which there isn't), it's result wouldn't be acceptable.
You can theoretically use the Player API to seek the video to whatever location you want, but this would be a major hack for a minor result.