I have heard some speculation that Tyrion catches Tywin in the act of strangling Shae and that prompts him to kill him. And I guess it would make some sense to do it like that to keep Tyrion a "pure" "hero" character in the eyes of the audience. But I feel like that's a cop out. Regardless of the reason Shae still betrayed him on some petty shit, and willfully lied and placed him in a position that he would have been killed, when he only tried to help her but she was too blind to see it. She has to go by his hand.