And I have never heard of carpool being used as a verb!
We use carpool as a verb in the US, so it sounds okay to me.
My issue with this is not the could/would thing. It's just not clear enough. When you say "Space is limited so please carpool so as many kids as possible can attend." to me, it doesn't read as asking some parents to bring multiple kids while other parents stay home. It suggests parking issues, but when you say the limit is 35 people, it's just confusing.
Here's the issue as I see it: You're basically asking a subset of adults not to attend, but to send their kids with other adults. Saying that straight out sounds just a tad presumptuous or rude, so you're trying to couch it in niceties by requesting that people carpool, but I think your real meaning is obscured in the polite translation. This may be one of those instances where you have to eschew the niceties and be direct, though.
I think cornflake's suggested wording comes pretty close, maybe with an added sentiment like "Parents, it's your day off! We are arranging carpooling for the kids, so we can have X pick up your child before the party and drop them off after."
How old are the kids? If this is a party for young kids, like under seven, I would give parents the option of accompanying their children if they prefer. For older kids, I think it's perfectly acceptable to say something like "You can drop your kids off at X time and pick them up at Y time."
Whatever you decide to do, I just think you need to make your expectations clear.