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On the other hand, if you create a parody of someone else's song, like Weird Al, you normally don't need a mechanical license as they are usually covered under the fair use act. Al apparently goes out of his way to clear it with the original artist anyway.

There is some confusion in the definition of a "mechanical license."  The term "mechanical" originally was meant to indicate that the sound had been recorded in some kind of physical format playable on a machine, as opposed to a notation like a musical score or a public performance of the work. Current law refers to a similar concept as a "phonorecord," and that is what most of us would call a recording, including digital information stored in a computer or a file download as well as CD, audio tape, vinyl records etc. In 1909 when the term mechanical license entered law, the machine was the player piano, and copyrighted music at the time is what we would call sheet music. So a mechanical license was intended to allow someone to make a mechanical rendition of the notated music, or subsequently more broadly to record a performance of the underlying musical composition (a term which now covers music and lyrics if applicable). Sometimes the term "mechanical license" is used to mean what the law now calls a compulsory license, under which you can record a performance of a published song regardless of the author's assent, because Congress in those days did not want a single company to be able to monopolize all such transformations and so made the granting of the mechanical license mandatory on payment of a set fee. 
 
When Harry Fox Agency sells what they still call a mechanical license, it is a license to produce and copy phonorecords of a new performance using the protected musical composition rights in the original song. If you create a defensible parody of someone else's song, a "mechanical license" bought from Harry Fox Agency would not be applicable, since it only permits you to make recordings of the protected musical composition rights in the original song. A parody necessarily makes significant changes to the original composition by changing words and/or music, and so it is the creation of a new derivative work, not a performance of the originally protected work. If Weird Al wants to get permission to do a parody, or if anyone wants to make significant changes in the original song, he needs to obtain a specific license to make a derivative work from the author or more commonly the publisher to whom the author has assigned his copyright.