Is it illegal to make a parody of a song then upload it on YouTube without the person who wrote the song's permission?
No. The court case that said it is ok involved 2 live crew, Roy Orbison and the song Pretty Woman. Google it; it should be available.
Long as you put the original author in the description as a disclaimer, you should be fine
Depends dont Mess with the big corporations though...
As long as you give credit to the original song you are fine