Right; however, with a parody of the sheet music that you recorded yourself you have fair use on the sheet music and no concerns about the recording. If you want to do a parody of the recording from the substantively the same sheet music, you may or may not be able to get by with nothing, but you would almost certainly be ok with a mechanical license, which the studio is required to give you for a known (small) fee.

In comparison, if you want to do a parody of a music video--a work that is considered to be on top of the music piece, as opposed to included with it, and thereby requiring a special "synchronization license" to make--using the original recording (which obviously implies the original sheet music), you are then going to have to negotiate with the studio, and they may simply not have any internal mechanism by which you can license it at all (even if individuals there think it makes sense).