I'm sorry, I'm not quite following you here. I understand your sister is acting like a bitch, but how about your parents? You said they don't invite your GF, but they invite you, right? They don't say "don't bring your GF with you" do they? From your post it doesn't seem to me they're doing anything wrong, but of course there could be some other things you didn't mention in your post, so sorry if I misunderstood.

Anyway, what I think is you should talk to your parents alone at first. Don't involve your GF and sister in the discussion, at least not at first. Just ask them do they not like your GF, and why not. They might hold back if she is actually there, and if they don't hold back and she's there she might be pretty hurt. If you get things sorted out with your parents, you could ask them to talk to your sister, it would probably mean more coming from them than from you since it's YOUR GF she doesn't like and she might just ignore what you say because of that.