Wow, interesting question. Hello, and by the way, this is not so "high school" as you think, in fact I think this is a real issue and your emotions are very real! So, don't discount how you are feeling. This is a tough one!
I am curious how you landed in this play group? Was it already established when someone let you know about it? Are these ladies friends from school who have a history together and need to learn how to let others into their group? Is someone "in charge" and decides when/where to meet and who else can come? Are you permitted to invite others?
A weekly group of 10 moms plus 10+ kids is larger than I can personally imagine, but I believe it is possible that you can cultivate one or two personal, closer friendships from among the women in the group. Is there someone that really stands out? Can you invite that person, and perhaps a friend from somewhere else (like a neighbor, colleague, sister) to a new gathering of just you 3 moms plus kids? If that mini-group works out, invite another to join. Start a smaller version yourself on days in which the large one does not meet.
Perhaps start a conversation with one or two moms about Brookfield Zoo (or wherever you have a membership or are interested in visiting) and say, "I was thinking of going there next Tuesday. Would you and your son/daughter like to meet us there?"
Once you feel like you have made an individual connection with a couple of the moms, see if anyone wants to meet up somewhere without the kids, like for a walk in the evening or an afternoon cup of tea, or whatever.
If you really enjoy the company of these moms, invite them to do something with you on a regular basis. It sounds like there is someone ruling the roost, choosing who gets to come to certain parties, and that is troubling me, since it is clear that the others, who are not invited, will hear about it later at the next play-date.
I suppose you could say, "Oh, sounds like that was fun night of cards! Let me know next time, I'd love to join you since I love poker!". If they invite you, go and have fun. If they talk about the others, be wary. If you are not invited, I would personally leave that group to find a new group of moms who are more inclusive of others.
Good luck!