“Dad Jokes” are a specific kind of joke, which typically make people groan, because they’re so terrible. See !dadjokes@lemmy.world, for example.

But his sentence could also mean that he tells jokes to his dad. She notices that and asks about it, therefore making a Dad Joke herself.
He doesn’t get the Dad Joke, which exposes him as perhaps not being very good at the whole Dad Joke thing.

I like to tell (my) dad jokes