It is important to make the distinction between parody that is derivative (as someone noted above) of another's creative work, which if violated, constitutes copyright infringement, and parody which is original, where offense is taking by the subject/victim of the parody, which suggests libel.

I think we've pretty well sussed out the copyright infringement question, which is easier. If it is a derivative of another song, it is a copyright issue. If the parody isn't derivative of a previous copyrighted work and the subject takes offense and sues, it will be for libel.

Here is an example of the latter:

http://www.nytimes.com/learning/teachers/featured_articles/20020520monday.html