I understand your point about dealing with parents in those types of work settings.
But, for example, big brothers big sisters-the parents sign their child up TO spend time with an adult man for example. It's the whole point that they need male volunteers.
Also, you could seek out friends with kids in the poly world too.
As I said, especially YOUNG 0-4 kids and maybe offer to babysit occasionally (aren't we all hard pressed to find GOOD, FUN, LOVING caregivers for our kids so we can get some extra time with our multiple partners?)
Or even an activity, if there is a group discussion and you aren't necessarily interested in the topic that will be going on, maybe offer that you would be willing to watch kids in another room, so polys who did want to participate-but wouldn't be able to because of young kid-could?
*I intentionally haven't elaborated on the other aspects of your topic-because others seem to have covered them well.
"Love As Thou Wilt"