We don't get to see much of Billy's father. We know pretty much four things about him: (1) He is a barber. That's why Billy's mother is so proud when Billy finishes optometry school and marries the boss's daughter; it's an economic move up for the Pilgrim family. (2) He throws Billy into the deep end of the YMCA pool to teach him how to swim, and Billy almost drowns. When he comes unstuck in time for the first time, in 1944, this is the moment he travels to, when he first seems to experience death. (3) Billy's father also freaks Billy out by taking him to the edge of the Grand Canyon and making him look down. And (4), Billy's father dies in a hunting accident while Billy is on maneuvers in the army.
All of this throwing-Billy-into-the-deep-end stuff makes us think that Billy's dad is supposed to be more of a manly man than Billy (though that isn't exactly a high hurdle). Billy has not learned much from his father in any obvious ways – unlike Roland Weary, who inherits all of his cruelty from his father. The narrator's dad seems somewhat in-between these two. Like Billy's dad, he a hunting enthusiast, but he is also a kind man. The narrator can take what his father has left him, his gun collection, and put it aside while understanding and respecting what it meant to his father.