Paul's father may not have physically abused him (at least, it is never mentioned in the story), however, he does intimidate Paul to such an extent that it would not be a hard guess to make that he does abuse him, at the very least verbally and emotionally, if not physically.
The main reason to suspect something is going on is that Paul is so afraid to go home after coming home from the theater late, on top of not coming home between school and work. The only image Paul sees is that of his father awaiting him at the top of the stairs ready to interrogate him as to where he has been. Also, it seems that if he had told his father what happened, his father either wouldn't have believed him, or wouldn't have understood.
Paul decides that he would much rather spend the night in the cellar of his house than go inside and face his father. While in the cellar, Paul fills his head with imaginations of his father hearing him sneak into the cellar and thinking Paul was a burgler. He then imagines his father shooting him without knowing that it was Paul. Next comes a scenario in which Paul screams out before his father can squeeze off the first shot and his father sees the error of his ways because he almost shot his boy. Finally, the one scene from this whole section of the story that worries me the most. Paul wonders if, later on down the road, if the scenario had taken place, would his father regret NOT having fired on him? It is this thought that Paul wrestles with until the morning light. It makes me wonder if Paul's father had ever told Paul that he loved him, or maybe he just tears into Paul all the time, for every mistake he makes (which could be attributed to a learning disability, most of which were highly misunderstood at the time of the story).
Paul's father also takes him out of school to send him to work, just because Paul's teenage mouth got a bit out of control. This is not something a caring nurturing father would do.
I have a kind of kinship with Paul, to the effect that both of us were raised by our fathers. I believe Paul's mother died, because in the setting of the story it was unusual for a woman to run off and leave her husband with the children. My mother also died, and for a long time my father was angry at the world and my younger brother and I (mostly me) caught a lot of that anger. I grew up in a semi-abusive home, but I was one of the lucky ones. Fortunately, my father got help before it became too late. My father is, however, a wonderful man, giving, caring, loving, generous to a fault. My father never emotionally abused me to the point that Paul's father has driven him. My father did an excellent job of raising two young boys largely on his own for nearly 10 years. It seems to me that Paul's father never had any idea what to do with the children (it is mentioned in the story that Paul has sisters) he was left with when his wife died.
It is because of Paul's intense fear of his father and his house that
I think Paul had been abused since he was a small child.