Bailey: I’m avoiding using someone’s preferred pronouns because I don’t respect their gender identity.
Motte: “What so using someone’s name is offensive now?”
From what they did say I think it’s reasonably safe to assume that the person they we’re talking about was non-binary, agender, and went by they/them but it really doesn’t matter.
A very common experience that trans people face is people unnaturally avoiding the use of their pronouns and either structuring sentences awkwardly or by using their name all the time. The literal act of using someone’s name isn’t offensive but when it’s done because you’re avoiding using their pronouns then it’s rude. And it’s super obvious when people do it which makes everyone feel uncomfortable.
Now because trans people deal with this all the time it’s possible that they were more sensitive to it than normal and overreacted but given the parent took offense to being called out and they used dismissive scare quotes to describe the person’s gender identity rather than it being a two second “oh my bad” thing I think suggests not.