No, Joker wasn't the one who killed Batman's parents. The culprit was a common criminal during a mugging incident. However, Joker has often taunted and tormented Batman about this tragic event in their confrontations.
In most mainstream comic continuities, the Joker didn't kill Batman's parents. It was a random act of crime by ordinary criminals. But there might be alternate or non-canon versions where this could be different.
There could be various reasons. Maybe he had mental health problems, or there was a major conflict within the family that pushed him over the edge. Whether it's true or not, it's a disturbing thought.
It could be Dumbledore from beyond the grave. He might find a way to reach out and summon Harry, believing that only Harry has the potential to truly defeat Voldemort in this new, dire situation.