Even in the comics, Batman figured out who killed his parents relatively early on. In a comic in the 1950s, Joe Chill is revealed to have been the thug who killed Thomas and Martha Wayne. Batman actually reveals himself to Joe, demasking and everything. Chill runs away to tell his goons in an attempt to get backup, but they get so pissed that Chill created Batman that they shoot him down in anger before getting to ask him who Batman was.

I'm not sure if any of this still holds in Post Crisis/Modern Era comics, but it wasn't just the movies that had that. Regardless, Batman still continued to fight crime.