Uncle Dad
A father who takes being the 'fun dad' so far that the role he plays in his children's life is more like an uncle. Do the kids need a trip to the dentist? Do they have homework? What are they having for dinner? He hasn't a clue - but he will occasionally take them out to a ball game. Often a non-custodial parent.
Oh hey, Uncle Dad sent me a birthday email and a $20 gift card to Baskin Robbins. I'm lactose intolerant and it's actually my sister's birthday, but... it's the thought that counts, I guess.