From the dialogue, David, at the point Kirk has the altercation with hin just outside the Genesis Cave, was unaware that Kirk was his father although he knew that his mother knew him. As he said on Regula I:

DAVID: Remember that overgrown Boy Scout you used to hang around with? That's exactly the kind of man...

CAROL: Listen, kiddo, Jim Kirk was many things, but he was never a Boy Scout!

Carol also doesn’t deny Kirk’s assumption when Kirk asks why she didn’t tell David about him. So it seems clear that David doesn’t know at that moment.

That being said, David may have met Kirk as a child. In the beta canon Kirk does spend time with David as a “friend” of his mother’s, but Carol insists that Kirk not tell David the truth, for the reasons she mentions in ST II about their different worlds.

It’s only after Spock’s funeral that David reveals he knows and is proud to be Kirk’s son. But the way this all makes sense is to assume that David was just told off-screen by Carol about his parentage.