Unfortunately, Bud never finds out who his father is.
While Bud spends the majority of the book believing that Herman E. Calloway is his father, it is revealed in Chapter Eighteen that Calloway is actually Bud's grandfather. After Bud sees that Calloway collects rocks and writes on them, Bud shows him that he has several rocks with writing on them, too. Calloway accuses Bud of stealing the rocks from the room that he had been sleeping in, but Bud tells him that he had received the rocks from his mother.
It is at this point in the story that Calloway realizes that he's never asked what Bud's mother's name is. Bud tells him the truth: Angela Janet Caldwell. Angela is actually the daughter of Calloway; she had run away from home because Calloway had been so hard on her growing up and had not been heard from since. Her death comes as an enormous shock to Calloway, who had been waiting for the eleven years since her disappearance for her to come home.
Bud only learns that his father could be a drummer from Calloway's band, a man his mother may have run off with. However, there is no definitive conclusion. Despite this, Bud finds himself in a new home and with a new family in the shape of Calloway and his band.