We don't get to see much of Billy's father... but we see enough not to like him all that much. 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) He freaks Billy out by taking him to the edge of the Grand Canyon and making him look down. And (4), he 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 macho macho man than Billy (though that isn't exactly a high hurdle). Billy has not learned much from his father in any obvious ways—unlike, say, Roland Weary, who inherits all of his cruelty from his father.