As a USC fan, this is easy. UCLA
They are obsessed with the Trojans. They exist just to hate USC. Most of the USC fans I know don't hate UCLA. We find them annoying, but we don't hate them.
They are the little brothers who want to go to the movies with you. If only they could win just one NC outright, or have more than one Heisman Trophy winner. Right now, they rank behind ND and whoever we play in a bowl game. We think about them one week out of the year, they never stop thinking about us. Kinda funny, really.