Such an event would qualify as a transformative work, in which the original art is changed in some way from the more famous version. It's determined on a whole host of factors, including the possibility for monetary damages, but you are allowed to, under certain conditions, profit off of derivative works. The question about monetary damages normally hinges on whether people are buy your work, believing that it is the plaintiff's, and that is costing the plantiff money.
One notable example is "Weird Al" Yankovic, who's music is generally humorous parodies of popular songs. While there is no legal obligation for him to do such, Weird Al famously gets permission from the original artist of the song because he is a stand up guy and is only doing the parody out of respect, and wants the artist to be in on the joke (Some more famous incidents including Michael Jackson denying his parody of "Black and White" despite giving approval to parodies of "Bad" and "Beat it", a rift between Al and Coolio after the former only got permission to Parody "Gangsta's Paradise" from the record company, not Coolio himself (who initially would have said no, but has since admitted he was impressed by Al's rapping), and Lady GaGa, a huge fan of Al, being absolutely mortified that the record company did say No to Al's request to Parody "Born this Way" without talking to her about it and called up Al to set the record straight). Al's just a nice guy, but he doesn't need to do this.
The reason that they claim copyright concerns is two fold. The "I'd like to tell that... ...but I can't" is creatively getting around that that they want you to be informed by what works to expect, without giving away specific songs that they want to do. If they don't want to do the Qween song where they confess to their moms that they killed a guy and are contimplating suicide because everyone does it and it's long, they can still "name" the band without giving away that they will be singing about their attraction to women with posteriors so large, they cause a rocky planet to spin.
It's also a humor in the vein of acknowledging that they are not the original song writers and most famous singers of the song (citing who is is a great way to get out of the money damage issues, because your making clear that you're not in any way, shape, or form, comparable to the most famous version of the song). Simpsons did a famous gag where, in a parody of "The Shining" they refered to the powers as "The Shinning" after Bart almost name drops the original work, prompting Groundskeeper Willie to shut him up with a protest "Ye want to get sued?!" Of course, this had not been the first time The Simpsons parodied a Horror Genera work... in fact, its usually their first episode of the Season... but it was mocking the fact that it was a famous story and we all know it.
And finally, someone singing a song they didn't write or originally perform. And infact some of the most famous songs of all time were written by someone, performed by a different person, and finally famously performed by who we attribute the song to. A great example of this is the sound "Hound Dog", which was written by a pair of song writers to be performed by an artist known as "Big Mama' Bertha". It didn't get famous until, as the writers tell the story, it was performed by "Some new guy named Elvis!" Disney's musical division famously released two versions of "Let It Go" by two different artists at the same time, believing the film version wouldn't be as popular as the pop rendition by a more widely known artist but feeling it would get radio play. Megan Trainer wrote songs for other artists before she became known. She was frustrated that her songs weren't getting picked up by famous recording artists and decided to just record them herself when they didn't sell.
This is called a "Cover" and happens all the time. The reason it is allowed is that music copyrights normally cover the unique vocals of someone singing and musical accompaniment. Under U.S. Copyright laws, words and prhases cannot be copyrighted so there is no law to stop someone from speaking those words in an artisticly unique way, or using a tune with different words (fun fact, Hum the tune of "My Country tis of Thee" and then hum "God Save the Queen") or taking the words from one song and singing them to the tune of another (another fun trick: Sing the first Pokemon Theme song to the tune of The Ballad of Gilligan's Isle). Or to a bunch of different songs (there's a cool video on Youtube of two guys having a genre sing off: They sing "Shape of You" in the styles of various genres and artists in full.