I've discussed setting up an event with expat families with young children with people on here but never followed through on it. The obvious place would be the Funarium, an indoor kid's play area near K-Village and the Rama 9 Big C, not right beside Sukhumvit but not far either. The idea was that the events here center on bars and such, but people with kids could let their kids meet other kids too, for people that tend to not do the bar scene.
In the hot season expats would tend to go there because it's inside, air conditioned, and I guess now because it's raining. The facilities are what one would expect: slides, a small climbing wall, trampoline, soccer area, ball pits, etc., set up in two zones for young kids and 4+ years. I really like parks better myself (Benjasiri isn't bad off Sukhumvit, and Rama 9 / Suan Luang is much better out on Naratiwat Road, by Paradise Park mall, way out there towards Bangna), or the Dusit zoo.
Post here if you have interest and maybe it will come together. They do events for birthdays, so a well-organized event could include a separate room, but something more informal would also work.