Well, generally a regular person will sue for defamation of character or invasion of privacy if they are parodied (a teacher sued a student for defamation over a MySpace parody done of him, as an example). They
could bring an infringement suit over a parody too, though, to protect a trademarked name (if they use it to market a product or service), or to protect something under their name that is copyrighted (think of a painter and his painting as an example).
Celebrities may have a fragrance or clothing line named after them, in which case a parody of them could also affect the marketability of any merchandise they sell. Someone doing a parody of, say, Britney Spears, could not only face a defamation action or an invasion of privacy action or a right to publicity claim, but also the parody could affect record sales or perfume sales, so the person may face a copyright and trademark infringement suit from Britney, as well. (Britney, I realize, is a bad example, as she is practically a walking-talking parody just on her own
).
Celebrities, however, as with any public person, have to meet a higher level of "proof" to bring a successful defamation suit or invasion of privacy suit. They must show that something was "published" with actual malice, which can be difficult to prove. An infringement on a celebrities right to publicity (everyone owns the right to their own image, and the marketing of that image, but celebrities images are more valuable) or a trademark or copyright action may be easier to bring and to prove than a defamation or invasion of privacy suit.
Since celebrities have voluntarily (for the most part) placed themselves in the public eye, they lose some of their privacy rights and they have opened themselves up to more comment and criticism (some of which can be rather scathing). What they have lost in
these rights, however, they have gained in their right to publicity. They can profit off their image, and this "image" and their name can be trademarked.
The
defense a person being sued can use, in all of the above instances, is that it was a "parody". Parody falls under fair use, both in copyright and trademark law, and under fair comment, in defamation and privacy law.