Arthur Fleck goes to the Wayne mansion to confront Thomas about being his dad but is, unsurprisingly, turned away. Not before meeting a young Bruce, though. When Arthur is eventually able to speak to Thomas, he tells him he knows about being his son, and Thomas tells him his mum is crazy and that she didn't even give birth to him — he's supposedly adopted.
In a flashback with Penny, she's seen being confronted by claims of child abuse over her adopted son, but she says the adoption was strategically fabricated as a coverup by the Wayne family. Because she's clearly insane, it's hard to say if that's true or not. So, presumably, we still don't actually know who Arthur's dad is.