From what you’ve said, I’d definitely move your dd if you are able to. Failing that, the school must address this. It sounds as though your dd is being bullied by the parents.
And who is telling you your dd is the only left out child? Surely these people’s kids would be ideal candidates for play dates?
The parents in my dd’s primary were particularly demanding and cliquey in comparison with some other years and other schools. I was never in the clique but not completely sidelined. Luckily a lot of parents did whole class parties in reception so that made us all interact with one another and I think it would have been a lot worse without it.
A couple of the kids with single mum younger parents were never invited to playdates, which I thought was really sad and I invited them to ours a few times over the years. Perhaps there are other kids, who are also pushed out by parents?
When dd was in yr1, her friendships dried up. With one parent there was a misunderstanding about something, which happened between her dd and mine and the mother freaked and closed down (nothing happened but she thought it did..long explanation required). Another mother also stopped play dates - fine for her dd to come to mine but wouldn’t reciprocate and with my very poor health, it was too hard. I got dd very busy with out of school activities. At first I sent her to rainbows as it was calm (and at the time only allowed biological women / girls). She soon got bored of this as she’s very active and I started her swimming, judo, riding lessons etc. Basically everything to ensure she was too busy for play dates. I also luckily have a friend, whose ds is the same age as my dd but in a different school so we also saw them once a week.
A couple of years on and we tentatively restarted play dates. Towards the end of secondary, the kids started to push more for what they wanted. Anyone, who came quickly learned our house was warm and welcoming. I fed them what they wanted (having checked with the parents) and was genuinely interested in the kids, lots of smiles and a tasty pudding, a fun activity or game but nothing desperate or ott. Slip and slide in the summer, trampoline. Dd is in yr9 now. I asked her the other day why our house is the place to congregate. Dd told me it was because of me always ready to make them feel welcome and offering food etc.
What I’m saying here is that just because you’re where you are right now, this doesn’t mean you’ll be stuck like this forever. It sounds as though you’ve got even bigger issues than us. But this isn’t surmountable. Sadly, the parents sound like arseholes. With the play date thing, you could suggest an alternative date with one or 2 parents. Unfortunately they could already have pegged you as needy. The playground is such a desolate place and the parents reenact their childhoods, if there’s one or even a few particularly controlling queen bees, you may be the unwitting victim and everyone is complying with them.
Don’t despair, find some activities your dd can attend regularly. Some can be very cheap. Judo is good for girls and send your dd to a childminder once a week if you can afford to do so as she will have kids to play with there. And her face will be a reminder to the parents that she exists.
Next birthday, if the budget allows, I would hold a full class party and also invite other children, family etc not at this school. Hire the local hall, bouncy castle and games. Ask any friends and family to help you entertain and clear up etc. Or hire an entertainer if it’s in budget.