This is not a common problem. Because when I paste a YouTube URL into a comment on card in Trello, sometimes it just ends with an embedded YouTube video which is just what I want, however, sometimes it won't work. The comment will show a plain hyperlink as I pasted in which is not good. How to solve this problem?
Trello uses embed.ly to generate previews for certain types of links, including links to youtube.
If you go to embed.ly, you can test it with that URL (http://www.youtube.com/twitterapi#p/u/7/HY5GIl7sFwE), and see that it isn't able to get a proper preview (because the link isn't in a format that it recognizes)
However, if you use a link like http://www.youtube.com/watch?v=HY5GIl7sFwE (which points to the same video), that is in a format that embed.ly recognizes, so Trello is able to get the preview.