All the parents in this show are truly awful human beings. Is this the same in the comic?
Not even remotely. The worst parenting I saw in the comics (I haven't read any rebooted stuff at all; I go by what's in the comic digests) was Coach Clayton. He's Chuck's father; Chuck is nothing like in the series - he's a nice guy, has a girlfriend named Nancy, plays sports, and wants to be an artist. But Coach Clayton is extra-hard on Chuck, since he tends to err on the side of emotional abuse rather than be perceived as being easy on Chuck because they're father and son. This is a fairly common problem when a teacher-student relationship is also a parent-child relationship.
Archie's parents are a bit old-fashioned, but they just want Archie to grow up and be a responsible adult. They see Veronica as spoiled and inconsiderate, and approve of Betty.
Betty's parents are both super-nice. Polly isn't in the teen comics; she's shown up in the "Little Archie" ones as a much-older sister, and by the time Betty is a teen, Polly is married with at least one kid. Polly and Jason never knew each other, and in fact Polly is much older. Betty's parents tolerate Archie, but disapprove of him for tending to treat Betty as second-best - that she's only good to date if his plans with Veronica fall through.
Jughead's parents went through a modernization during the time I was actively reading the comics. His mother was a frumpy housewife who was constantly nagging Jughead to do his chores (he had loafing down to a fine art), but she was eventually updated to become younger, not a frump, and Jughead now has a little sister called "Jellybean."
Reggie's mother is never seen in the comics I read, and never mentioned. His father owns the local newspaper, and while not as rich as the Lodges, the Mantles are upper middle class. Reggie evidently gets a very generous allowance, which explains why he's always got a flashy car, flashy clothes, and can take Veronica to expensive restaurants and concerts. We don't see much of Reggie's dad in the comics.
The Lodges are wealthy, and Veronica goes to Riverdale High because her parents don't want her to get
too spoiled. Mrs. Lodge of the comic digests is a white-haired, nose-in-the-air society matron, with clubs, teas, and a country club lifestyle. She's not seen very often. Mr. Lodge is a tough businessman (not a crook like in the series), and he tries to get Veronica interested in business, or at least working at some sort of normal teenage part time job so she can learn what it's like to earn her own money. He's very generous with Veronica, but as mentioned, doesn't want her to get
too spoiled. There have been times when he's told her to go get a job "like Betty does" if she wants money for a new dress. His issue with Archie isn't that Archie's parents are (relatively) poor; it's that Archie tends to be a klutz, breaks expensive stuff, and lacks the sort of manners necessary to mingle with the Lodges' socioeconomic class. He approves of Reggie, even though Archie is much more considerate, and honestly tries to be helpful to Mr. Lodge... it just never seems to quite work, though.
I've never seen Jason and Cheryl's parents in the comic digests. Ditto for Josie; in fact, there's never any indication that Josie and the Pussycats are even still in school. They seem to dash around doing concerts at the drop of a hat and travel pretty much anywhere, so I don't see how they could still be in school.
In the current rebooted comics, Jughead's family used to be super-rich (they were famous for a product with "Jug" in the name, which was how Jughead got his nickname in the new version), but they went bankrupt. I don't know if his parents were rich in the original comics, but for what it's worth, his real name is Forsythe Pendleton Jones III, which helps explain why he prefers to go by Jughead.
Jughead's parents in the original comics were middle-class.