At the end of Batman (the 1989 movie), Batman says to the Joker 'you killed my parents'; the Joker replies that he (Jack Napier) was a kid when he killed them.
Does this mean the Joker knew Batman is Bruce Wayne?
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It's possible, but not until the very end of the movie.
This is the script of the scene you are recalling (found here):
The Joker, hearing something beneath the trapdoor, begins to move the second bell over it. However, his curiosity gets the better of him and he lifts the trap a fraction of an inch. He opens the trap fully, no Batman.
JOKER I must have belled the bat!
Joker LAUGHS then stops and looks around slowly.
JOKER There ain't any more of you up here, are there? -- Daddy or Momma bat?
The Joker LAUGHS, CRAZILY.
BATMAN (O.S.) My parents are dead. But you remember that, don't you, Jack?
Joker turns quickly to see Batman standing with cape un- furled. A rappeling line hangs from his belt and out onto the parapet. A small pulley on the belt. The Joker CACKLES.
JOKER You climbed up the outside with a rope! (LAUGHTER) You little monkey! God, look at you. You're a mess. (laughs) They're gonna be real mad at the costume shop when they see what you did to their suit.
Batman steps toward him. Joker steps back into the half-dark.
BATMAN (wicked smile) Yeah, and they asked me to bring your face back to the shop for some work. Even if I have to tear it off. JOKER You maniac, it was you who dropped me in the tanks. You made me. BATMAN I made you. And you made me. JOKER What is this? I say you made me and you have to say I made you? How childish can you get? You're insane. BATMAN (the voice of doom) Are we going to kill each other, Jack? JOKER (O.S.) (slightly nervous) Jack? Jack's out, I'm running his body while he's gone. BATMAN Well, when you see him -- tell him I'm gonna kick his ass!
If we go by the evidence in this scene; Joker is unsure if Batman has backup, possibly consisting of a 'Daddy bat' or 'Mamma bat'. Therefor at this point he does not know the true identity of Batman. The next line, delivered by Batman, gives Joker a hint at Batman's true identity. At that time it is possible that the Joker was able to determine Batman's true identity, but it's never made clear that he does.
If you take just that line by itself, It's implied that Joker has in fact figured it out-though when this happens would probably not be until the fight in the Clock Tower itself. Being that Bruce survived the bullet from earlier in the film at Vale's apartment, and his tenacity going after Vale at the museum and then the clock tower could be enough of a stretch for Joker to have figured out that they are 1 and the same.
However, taking in the entire film-I would say this is just Joker saying his line based off of the fact that Batman is at least in his 20s, and the reference to "killed my parents" implies Batman was a kid at that time-could be matched to any number of his homicides from 15-25 years ago. Napier is at least 40 in the film, so the time would make sense.
Even if he had his suspicions, he never got a chance to use the knowledge, or even mention anything later.
BATMAN just said you killed my parents. Jack may have done lots of things but I'm sure he's READ about Bruce in the paper and knows about him and kept tabs of sorts. Don't forget he waives at Bruce in limo... So when bats says that line - I'm sure Joker instantly thinks OH SNAP it's Bruce Wayne! At that point it's only logical the millionaire kid has become batman - who else WOULD it be, right? Also Jack is like a genius would be serial killer etc... He's a pyscho and brilliant. I'm sure it wasn't hard for him to put that together quickly. Nicholson COULD have made a funny bit out of that - with a realization and funny line... They do gloss over that pretty quickly lol