Jughead Jones’s relationship with his dad is never perfect, but it always exists on a level far higher than people seem to think. FP Jones has made mistakes in his life -- in different verses, these amount to different things -- but at the heart of all his choices and all his errors is the love for his family, for his son, and Jughead knows this without fail. He may doubt the man at times, but there’s a connection that runs deep there. There is a great deal of respect and trust, even if it’s sometimes not entirely earned by his dad, or shown by him. At the end of the day, Jughead knows how much his dad cares about him and loves him, and chooses to believe in this always. Sometimes it’s easier than others -- he’s not beyond feeling abandoned, forgotten, left behind. He’s also not above judging his father, just as he judges his friends. But he never fully villainizes his dad the way the world seems to. He may blame him from time to time, force him to own up to his choices and address those consequences, but it comes from a place of true familial love. He wants to see his dad do better, he wants to see his family as a whole. He knows FP is capable of anything -- he’s proven himself to be smart, hard working, and driven enough to succeed, and he wants to see that for him.
In his life he’s always felt the burden of greater responsibility than most kids his age. Helping look after his little sister was always something that helped him grow up, and when financial troubles started in on the Jones family and the fighting began between his parents, he found himself mediating and stepping in, for their sakes as much as his sister’s. He became an observer to his family, an objector, and it made him more equal with his parents than the typical parent-son relationship he saw with his friends. It also made things trickier -- because his parents did still see him as the child in the situation. It was why he inevitably felt unable to keep his family together, though that was beyond his control.
In general he has a somewhat distant relationship with his mother. They’re close in many ways, but in part due to the way Jughead was cut off from their Chilean heritage and how Gladys unites with more of it later in life, he only really has pieces of that world that his sister has more of. He was much closer with her when he was younger, happily doted upon. As a teenager, the things they bond over most are food and literature, and those are the things that can draw them back together even during times of greater separation.
His sister he will always be protective of. They have a very deep connection that is virtually unbreakable. With how much of a hand he had in helping raise her, there’s very much a paternal instinct as well, felt by both of them. But the lines are blurred as they grow older, and though it will always be there, they are certainly equals. That isn’t to say that they don’t fight like siblings do. There are harsh judgements made about life choices on both their parts, and it isn’t always easy making up after a famous Jones blowout. But Jughead always does his best to make peace, feeling far better in all things when he and his sister are on good terms.
Back to FP ... There’s hardly a more important person in Jughead’s life than his dad. His dad becomes a role model in different ways for him. There’s much about him that he wants to avoid becoming, but much about him too that he hopes he has in him. His life hasn’t been perfectly happy, and his childhood wasn’t always stable -- but he found so much light in hard times, and those moments were always tied to his dad, first and foremost. Snow days as a kid, being picked up from school in the truck, sitting on the back of his bike trying to reach the handlebars, Independence Day, and everything that went in between. Jughead loves his dad, and despite all the negatives, he always has hope about the man that he is.