Covers are unique in that depending on how you are doing the cover which licenses you will need. If you're not using the original recording (such as an instrumental version of a song), then you would not need the mechanical licenses, but would still need the composition licenses. On top of this, you will need a license to grant you permission to perform the song, often considered "busking," and sync licenses for performing them on a stream.
Parody is challenging because for it to actually fall under "fair use," something only a judge can decide, the parody must actually parody the song itself or the artist. Songs about another topic set to the same tune or music are not protected under fair use for parodies.
Remember, fair use is decided in court on a case-by-case basis. Fair use will not stop you from receiving a DMCA take-down notice. However, if your content does indeed fall under fair use, it will protect you and your counterclaim. (You can read more about counterclaims and the DMCA process here.)