I’m not quite sure when this particular piece of functionality was added to YouTube, but I do remember writing a tweet about how desirable it would be to have a url that started playback at a particular point in a video.

I’m still not sure how to get it to work with an embedded video (such as the wonderful Tim Minchin video above), but it certainly seems to work with a direct link (which will take you directly to the beginning of the song - note the #t=0m37s at the end of the URL).