How do you parody an instrumental?
Is your arrangement for the kazoo and the calliope withe whoopie cushion noises? Is it a marching band tune that you are scoring for the piccolo and the mouth harp for comedic effect?
Remember that the lyrics and the music are two independent works as well as a joint work and while it is relatively easy to parody the lyrics, and possible to parody both, it is much harder to parody the music on its own.
In any event, it seems that what you are talking about is satire, not parody. You are making fun of relationships: that’s satire. To be a parody, you have to be making fun of the song. Parody is fair use, satire isn’t.
In fact, when examining how copyright law protects parody the Supreme Court explained in Campbell v. Acuff-Rose Music, Inc., “Parody needs to mimic an original to make its point, and so has some claim to use the creation of its victim’s (or collective victims’) imagination, whereas satire can stand on its own two feet and so requires justification for the very act of borrowing.”
If your critique of relationships would be just as valid set to other music, you don’t have a fair use defence.
Note that what is protected is “criticism or comment” which can be a humorous parody but can also be serious. The guide is not if it’s funny, it’s if it criticises or comments on the original work.