You are aware that Batman is a fictional character, and therefore neither he nor his parents actually exist aren't you? So, from that, nobody killed Batman's parents because there is no such person.
However, if you mean who originally killed Bruce Wayne's parents the first time that the character was created and that part of his life was first invented, it was a random un-named mugger in DC comics #33. This became Joe Chill in Batman #47 - and in DC #235 it turns out he wasn't a mugger but a hitman hired by a Mafia boss called Lew Moxon.
In the movies, Batman (1989) showed Jack Napier (the Joker) as being the murderer, whilst Batman Begins goes back to the Joe Chill story.