A cover is when you just play the song yourself. That's section 115 of federal copyright law and there's rules for it.
A parody is when you deconstruct something to mock it and the deconstruction is financially not competing with the original. It's why Weird Al always asked permission, because although "Eat It" had silly lyrics, if Michael Jackson cared he could have argued it sounded enough like "Beat It" to be financially competitive in the music market (and let's be honest, some people did accept the Weird Al versions as substitutes. I know I prefer "The Saga Begins" to the real song), and Weird Al might have had to go to court and argue his case. If he lost, he'd be forced to pay royalties, likely following the rules for a cover song.
Parody is section 117 and it's a lot weaker than people think it is. It's just that court is stupid expensive and nobody wants to go to court.