Not true. Parody is not protected against by copyright so much. This is a grotesque oversimplification, but basically, if your making fun of something and not hurting the original in any way except to make fun of, you can't be stopped. It's called, "Fair Use" if I remember correctly. Weird Al has every right to release every one of his songs without asking permission. He just does out of respect.
Also, I know that plenty of people want to crucify someone who they don't like the music of because they are to classy for that lowbrow pop music, but seriously? Weird Al, particularly thoughtfully, said himself that he talked with the manager, not Gaga personally. Like her or hate her, who do you think is more likely to be a stickler for use of her song, Gaga or the buisnessman watching her Ip's? The possibility that the manager said no without Gagas knowledge is absolutely, completely believable, and there is absolutely no proof otherwise. Maybe Gaga was behind this, and then spreading blame. I can't say one way or another. But most importantly, neither can the people getting in arms against Gaga. Weird Al is awesome, he is a stand up guy who has a great amount of respect, and rock solid principles. But I would bet that he is clashing with a manager, not Gaga herself. When someone has evidence to the contrary, they can argue against my hypothesis.