Bruno's dad is one of those characters whose presence looms throughout the text, despite the fact that on a page-to-page basis he's a pretty minor character. After all, it's his promotion that the family moves for. He's the guy working for Hitler, fostering a home life for our main character devoid of criticism of the Nazis. Bruno's mom may keep her opinions to herself, but we're thinking she only does so because to do otherwise would be to defy his father. This guy, it seems, runs the show. At one point he says to his son:
"Do you think that I would have made such a success of my life if I hadn't learned when to argue and when to keep my mouth shut and follow orders? Well, Bruno? Do you?" (5.262)
When he says this, it's super clear what kind of household this dude heads—he definitely expects his son to fall in line and keep his questions to himself. Which, of course, is exactly what we see Bruno do (more on this over on his page elsewhere in this section). Bruno's dad is pretty much what we'd expect from a Nazi commandant: strict, cold, businesslike, and powerful. In other words, a force not to be reckoned with by his wife or children.
Like many Germans, Bruno's father believes in the Final solution because he sees Jewish people as less than human. He explains:
"They're nothing to do with you. You have nothing whatsoever in common with them." (5.284)
It's a total refusal of shared humanity. But if you read up on Bruno and Shmuel elsewhere in this section, it's crystal clear just how wrong Bruno's dad is.