Weird Al himself released his Lady Gaga parody online without permission. He thought he had permission when he made it, then, when he got the runaround, he went ahead and put it online. He’s since gotten permission straight from Gaga herself, who blamed the whole thing on her agent or similar.
Given that, and the huge number of parodies on YouTube without permission, I wonder if the information about fair use is current.
I’m not “objecting” to your idea, I’m saying that it is deeply naive. Parodies of TV theme songs is probably the lowest and easiest form of filking. Write them down, I’m sure you can google up fora where people share such things and might get the occasional “lol” out of it. But nobody, and I mean nobody is going to pay you for it.