I don't remember where it's mentioned in the books but someone says that the mad king tried to fuck Tywin's wife a bunch of times so that's where the Tyrion being a Targaryan thing comes from. Plus I think one of his eyes might be purple(?) and Tywin's always hated him
One eye is black.
There's a small remark from Barristan Selmy to Daenerys:
“Prince Aerys . . . as a youth, he was taken with a certain lady of Casterly Rock, a cousin of Tywin Lannister. When she and Tywin wed, your father drank too much wine at the wedding feast and was heard to say that it was a great pity that the lord’s right to the first night had been abolished. A drunken jape, no more, but Tywin Lannister was not a man to forget such words, or the . . . the liberties your father took during the bedding.”
The fan theory has to take this as fact and literal, which as we know different stories from different people can describe certain events completely different.
Tywin as we know truly loved his wife so it doesn't make sense to me he'd be like, ah fuck it you raped my wife we cool bro. As someone else wrote:
Why did Tywin continue to serve as Aerys’s Hand if he suspected this? So far as Tywin or anyone else ever indicates, his departure from the role of Hand and his fateful break from Aerys were the result of long-building tensions thanks to the king’s madness and his stealing away of Ser Jaime to serve in the Kingsguard. For that matter, why did he persist in trying to wed his daughter to Aerys’s heir, if he had cause to hate Aerys so much?
I think people want to believe Tyrion is a dragon rider but people overlook the fact that you don't have to be a Targaryen to be a dragon rider, they were only a small minor house in Valyria and presumably anyone can be a dragon rider (and that's not even getting into the dragon horn).