How did The Joker know who Batman's parents were?

Joker used to be a big time mobster Jack Napier. These don't usually run around killing kids' parents (or, if they do, kids are unlikely to survive that). So, it's not hard for him to realize that this must have happened when he was a thug, i.e., when he was young.

Notice how he first asks "What are you talking about?" and only after that says that he was young. I don't think he knows who are these parents, but does conclude that it happened when he was young. After all, he's crazy, but very far from stupid.

Maybe Joker didn't know

Joker's initial response is "What?" before he laughs it off. At least at this point, it seems he genuinely doesn't know what Batman is talking about (specifically), the information is news to him.

Joker spends the rest of the conversation dodging the topic, instead focusing on reasons for Batman to not attack him. This is clear by the "not hitting a guy with glasses" joke.

It's perfectly possible for Joker's "I was a kid" argument to be fictituous (just like the "wearing glasses" argument) because they are both common reason to urge Batman to stop his aggression towards the Joker.

As much as he acts the fool, Joker is not an idiot. He's exceedingly capable of reading people and situations in order to remain unpredictable. His ability to taunt and defy Batman is unrivaled, and it stands to reason that his abilities also make him able to pretend to know what he's talking about ("I was a kid back then") when in reality it is an educated guess, not something he knows.

Maybe Joker did figure it out then and there

Joker's initial response is "What?". This sounds like a genuine response of confusion.

Compared this to the second response, "What are you talking about?", which only comes after Joker has had time to think of a response (stalling the conversation by laughing to get some time to think before actually responding).

So what could he need the extra time for by stalling? What could he be thinking about? Look at what Joker knows:

• Batman has considerable resources at his disposal as proven by his activities.
• Based on Batmans' deep resentment for Joker, it's most likely that Joker didn't recently kill Batman's elderly parents, but rather killed them when Batman was much younger
• Given Thomas Wayne's stature, Joker would definitely remember killing him (if anything, he would revel in upsetting the upper echelons of Gotham's society, which killing Thomas Wayne would definitely achieve). If you asked Joker "which rich guy did you kill", Thomas Wayne would likely be at the top of his list.
• It's highly unlikely that the Joker has killed many married couples at the same time.

As much as he acts the fool, Joker is not an idiot. He's exceedingly capable of reading people and situations in order to remain unpredictable. His ability to taunt and defy Batman is unrivaled. It's perfectly possible that he's able to make an educated guess as to who Batman could be; based on who he's killed, Batman's implied wealth and deeply ingrained resentment for the Joker.

Joker is exceedingly able to remain unpredictable

He is the living embodiment of the chaos element, and makes it his life's work to taunt and ridicule Batman. His behavior is carefully crafted to be all over the place, making it impossible to predict his behavior. This scene you've linked is no different.

This makes it very hard to figure out if what Joker says to Batman is truth or lie. His ability to read Batman means that when Batman will assume he's lying, he'll tell the truth, and vice versa. The more you can argue that it must be a lie or the truth, the more likely that it becomes that it's in fact the opposite.

I actually really enjoy this scene because we can't know for sure what Joker does and doesn't know. It's exactly how his character fits in the greater narrative.