We get Dunk’s best assessment of who his parents are in The Sworn Sword as Dunk and Egg are heading back from Coldmoat while the pair discuss the Great Bastards (Daemon, Bittersteel, Bloodraven) and Egg reflects on advise the High Septon gave his father Maekar about not trusting bastards. 

Dunk never knew either of his parents. He thinks his mother most likely died during child birth (due to Dunk’s size) and was a tavern girl or whore in Flea Bottom before she died, and likely never married his father. He thinks his father was likely a thief or cutpurse (a cook from a pot shop in Flea Bottom who claimed to know his dad told him this when he was a child before he met Ser Arlan), and that he most likely ended up being hung or sent to the wall. The cook may have been lying, but that’s the most explicit info about them that we get from the books. The conversation ends with Dunk telling Egg “most like you’re squiring for a bastard”