Filks and parody

My understanding of "parody" as applied to copyright is not the commonly held understanding. Not every comedic or critical use of a work is parody for purposes of copyright! Parody can constitute fair use, but for it to be fair use, it needs to make fun of (or otherwise comment on) the work being modified. So creating alternate lyrics to Nirvana's "Smells like teen spirit" that make fun of the president would generally not be fair use, even if it is, more generally, parody. To parody a song you have to be making a statement about that song itself, or it's not fair use.

Am I right? If you disagree, please provide links to support your argument. In the meantime, Family Guy is getting sued for modifying "When you wish upon a star."