Hi.
The way I've always thought it'd go is to contact the party that owns the rights to the tune. Just like when You're recording it as a cover.
They, who ever they might be, will then either give the permission for a fee, or not.
I doubt that he will answer, but Weird Al Yankowiz might know a thing or a two about the legal issues around parodies

. Perhaps Googling legal issues about his art would yield some results.
BTW, it's really hard to imagine that Your library or county has got zero lawyers who could give you the up to date answer according to your states laws.
Regards
Sam