In the character of Father we have a highly representational figure, much like his wife Mother. Seriously, we want to hear those wedding vows: "Father, do you take Mother to be your lawfully wedded wife?" Oh, no, that just sounds yucky.
Father's biggest role in the novel is to show a man stuck in the 19th century, unwilling to change and not wanting to. He likes things the way they were—with his wife obeying him and "colored" people acting the way they should. Even when he takes his son to a professional baseball game, he becomes annoyed at the way the players swear, remembering instead when he played at Harvard and the game was played "avidly, but as sportsmen" (30.4).
Yep, he's a throwback. But if there's one thing we can admire about Father it's that he has a sense of duty. He tries to help Coalhouse at first when his car is vandalized. He then negotiates with Coalhouse in an attempt to stave off the violence and save Coalhouse's life. And, as the U.S. prepares for World War I, he feels a strong duty to country and helps out the cause.
But he still feels lost. He feels born into the wrong era. All these newfangled inventions startle him, and he's weirded out by the new ways of thinking that have become the norm. He's like a grandpa in a rocking chair lamenting about the way things were in his day… except he's not old, he's a middle-aged man, and should have enough life in him to change with the times.
It's not going to happen, though. We even get the sense that he even liked it better when his wife wasn't that into having sexytimes, because it was right and proper for a woman to be icked out by sex.
On his voyage to the Arctic, Father gives into his repressed sexuality and sleeps with an "Esquimo" woman, but then feels strong guilt about it on his return. He's moral, but he's totally inflexible. When he gets back from the Arctic and finds his wife is interested in gettin' it on, he's shocked… but also pretty excited.
His hypocrisy about sex mirrors his hypocrisy about race. He supports Coalhouse, and yet at the same time wants him to act more humbly. Coalhouse's pride and determination are attributes that Father supports, but he still thinks that it's only right and proper that a black man should act meek and subservient to white men.
Basically, Father is a man who doesn't know himself, a man who is perpetually unhappy, a man who is "disturbed by his nostalgia" (30.4). He symbolizes the patriarchy that's challenged by the new century and all of its newfangled inventions and freedoms.