While only a court can rule on whether something is Fair Use or not, Loudr (who handles our cover licenses) offers the following guidelines:
If a parody takes aim at the original work itself, it is typically considered a true parody and is likely to be Fair Use. An example would be Weird Al's "Achey Breaky Song", which parodies the structure and criticizes the repetitive nature of the original song "Achey Breaky Heart".
If a parody simply involves a change of genre or lyrics and does not provide critical commentary on the original work, it might not be considered a Fair Use of that work. An example would be Weird Al's "Eat It", which puts parody lyrics to Michael Jackson's "Beat It". However, the lyrics are about food, not about the original song itself.
If an artist intends to make a cover song (not a parody or satire version) where the genre and lyrics were fundamentally changed from the original composition, then it would be considered a derivative work. Derivative works require permission directly from music publishers, which a mechanical license secured through Harry Fox Agency would not protect.