Is there a phrase or word that would fit the activity (verb or noun) of hosting or supporting an activity for your children etc. Such as coaching the soccerteam, or perhaps just taking them to a party/sport/activity and staying there with them.
(noun) : an older person who accompanies young people at a social gathering to ensure proper behavior; broadly : one delegated to ensure proper behavior
2 to act as chaperone to or for
to act as a chaperone
Chaperone can be used as a noun or verb. You could say, "he's a chaperone", or "she's chaperoning the activity"
Anthropologists use the term participant observation for their ethnographic research; though the term does not exist, perhaps you're looking for something along the lines of participant parenting that communicates active parenting and is easily understood in context.