We all know that Tyrion killed his father with a crossbow (who disappointed us all and didn’t even shit gold in the end), but is it possible that Tywin would have died anyway, even if Tyrion never stopped at his father’s room during his escape?
Most people, myself included, found it odd that Tywin Lannister was rather untouched by Oberyn Martell’s infamous fury in A Storm of Swords. When the Red Viper first entered the city, he made it very clear that he would have justice for Elia and her children.
“I have hungered for a long time. Though not for food. Pray tell me, when will the justice be served?”
A Storm of Swords, Chapter 39; Tyrion V.
He did challenge the Mountain in a trial by combat in Tyrion’s trial, hence managed to annoy Tywin and Cersei and got a chance to kill the Mountain without any room for open complaint from the Lannisters (trial by combat is a “holy thing” after all). But what good is killing the piece when you leave the player be? Tyrion himself later realises that Oberyn doesn’t only mean the Mountain’s death when he says he hungers for justice. (”He wants the head that spoke the words, not just the hand that swung the sword.”)