A Parody (caps) in legal terms is an artistic critique, commentary or exaggeration done to make fun of a piece or artist by using source material related to that piece or artist. A solid example would be the song Smells Like Nirvana by Weird Al Yankovic. He uses the composition of the song Smells Like Teen Spirit to make fun of Nirvana. This is the true essence of Parody in US copyright law and is protected under the Fair Use provisions.
A Satire is similar to the above, but contains no connection between the source material used and the target of the piece. To stick with the Weird Al example, this would be like his song Amish Paradise which uses a Coolio song (Gangsta's Paradise) to make fun of the Amish. Satire is not protected under Fair Use.
The tricky part of the above is that YouTube runs automated matching systems which do not consider Fair Use, so while doing a Parody and using original source material may be legal, it is not practical as the YouTube ContentID system will still flag it.
Thus, when doing a Parody, you need to record your own backing track in addition to everything else in order to be safe as that backing track should be different enough so as to not be automatched by ContentID.
Essentially, your argument for fair use is that you must use or reference copyrighted material in order to make fun of said copyrighted material. This is the logical progression of a Fair Use Parody defense. It's also why Satire doesn't work, because there is no logical reason why you needed to use a Justin Bieber song to make fun of President Obama, for example as you could have picked any song. There is no Parody license, so to speak because Parody doesn't require one. It is possible to request or acquire a synchronization license for this purpose, or artist permission as Weird Al always does, but it is not required.
http://fairuse.stanford.edu/overview/fair-use/what-is-fair-use/
I should probably write a sticky submission for this, as I've typed this explanation out probably 10 different times now.
