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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?

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.

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.