Get real — why
would they decide they were against LGBT rights? What if there were a good reason? — I can't think of one but we have to assume there'd be something that caused them to make that decision. If they had a strong case I might have to reconsider my views on the subject. And if they didn't have a good reason don't you think the LGBT community and its many friends would create a friendly platform for them pretty quickly or attract high-powered lawyers who'd argue their case if litigation was a choice?
For the most part, the LGBT community as a whole probably has no difficulty adhering to the user agreements they have with their hosts. All Parler had to do was exercise a little bit of content moderation, as is done here and on many other message boards.
The alt-right's big problem here (and Trump's) is that they have a lot of angry, violent people in their user base and these people would leave the site if they felt they were being muzzled. They want a place where they can swagger, threaten, and vent.
Fomenting violence and advocating insurrection isn't a characteristic of LGBT culture. On the other hand, it's in the political DNA of many Trump followers. They need to learn how to express themselves in a way which comports with the terms of service of their hosts or find friendlier platforms.
It has nothing to do with the 1st Amendment.