Only blue and green dragons would systematically choose gay riders, and those were largely considered disposable - although a few appeared in the narrative, most often they were only mentioned in the stats of how many of each colour hatched and how many died. Gold, bronze, and brown dragons wouldn't choose gay riders, and riders of those dragons were the only ones eligible for any leadership positions. So by default, gay riders were automatically and permanently relegated to subordinate positions due to their sexuality. When you combine dragons being picky about their rider's sexuality while also insisting that each colour of dragon prefers certain personality types, you end up with some really unfortunate implications. The best thing the Pern fandom did for the fandom was decide that dragons don't care about someone's sexuality, severing the implied link between sexuality and competence (since green dragons were persistently described as being flighty, unreliable, and dramatic.)

And the classist elements are there all through the entire series. It's supposed to be some amazing feminist story because only women can ride gold dragons, and gold dragons are the highest ranking ones. But they're also only a tiny handful of all dragons, less than 1%, and all the other women on Pern are largely treated like shit because they're only seen as breeders for the next generation. Conveniently, most gold riders also seemed to come from particular bloodlines, with it being noted that Ruatha Hold in particular provided a lot of them. So basically the message here is that a small portion of women, if they come from sufficiently good bloodlines, are worthy of being treated with respect, but none of the others are.

Pern is absolutely classist as hell, and any stories about characters from lower classes always inevitably end up with them becoming a higher class, whether that's Impressing a dragon (preferably bronze or gold), joining a crafthall (preferably the Harpers, given that being a musician, archivist, or spy has more "prestige" than being a tanner or blacksmith), or being given land in the Southern Continent to become members of the aristocracy themselves. So one could argue that there's an aspirational element, but "lower class person can only achieve anything of note if the upper class give them the opportunity to become upper class" is still pretty classist, especially given that with the dragons, it's related to a genetic trait and also requires a specific combination of sexuality and personality.

There were also specific rules about what colour Pernese people had to be, but because this simultaneously made blonde haired and blue eyed people incredibly rare as well as black people, you can argue the toss about whether Anne McCaffrey was also a racist. Obviously by modern standards, "all the human races interbred so much that everybody averaged out to looking kind of Latino" is not a great look, but it was common in a lot of sci-fi of the 70s as a means of implying that everybody stopped caring about race. But the Ruatha Hold family were specifically established in one of the books as founded by red haired blue eyed people, so Anne McCaffrey's most favoured family line on Pern also happened to be really, really white...

And the problem is that it's not just that the world was like this, because obviously what an author writes is different to what they believe. But the portions of the fandom aged 35+, who were active in the fandom in the early days of the internet, can confirm that Anne McCaffrey actually believed it, and regularly attacked fans who didn't agree with her, either on her forums or through cease and desist letters from her lawyers. (I got one. I was 16.)

Don't get me wrong, I do love Pern as a setting. I still dabble occasionally, but only because when Anne died, Todd McCaffrey did away with sending threatening letters to any fan who stepped out of line, and the fandom as a whole decided to make Pern less homophobic, less misogynistic, less classist, and less racist. I feel like fans today, who have only experienced the adjustments fandom made to Pern, have no real idea of just how toxic it was in the past.