The big, bad dad. He's even got the hairy legs of a fairy-tale beast. What's not at all clear is why, exactly, Paul's father is so bad. Is he abusive? Not that we see. Is he shiftless? Nope, he's apparently a hard-working, stand-up guy, with a "worthy ambition to come up in the world" (1). Is he just grumpy and mean? On the contrary, he gives Paul money to go study geometry and is even occasionally "jovial" (1).
The problem seems to be that Paul's dad is fundamentally a conformist. He does exactly what he's supposed to, according to the norms of his community—and admittedly his community has some pretty dull norms. He works for a big corporation; he sits out on the steps every Sunday; he approves of boys earning a little money; and he wants his son to follow in his footsteps. This is hardly the stuff that memoirs are made of.
At the same time, it's obviously a living death to Paul. He is so afraid of becoming like his father that he's willing to die to prevent it—and die he does. So is Paul's father indirectly (or even directly?) responsible for his son's untimely end? Or is he just doing the best he can to raise his kid?
As if dad weren't bad enough, Paul has a whole passel of authority figures after him. These teachers have it in for Paul—bad. They can't even talk about him without "rancor and aggrievedness" (1.3). With the way they treat him, it's really no wonder that he has a "hysterically defiant manner" (1.3).
It's not clear who started it, but we'd have to say that the blame probably lies on both sides. Sure, Paul is insolent and defiant. But—in his mind—the teachers are forcing him into a mold that he feels is literally going to kill him. In that way, they're the very definition of antagonists.