@japchap You're only seeing a slice of the time that family spends together.
At home, the parents are part of the kids' playthings, because everything else quickly becomes static and boring and too well-known at home to hold the kids' interest.
Out at the park, the kids finally have new experiences to take in, relieving you of being the entertainment for a short hour.
The kids don't need to go to the park to experience their parents; they go to the park to experience the park and all it has to offer. The parents are just a safety net while they explore.
Or you get the helicopter parents who micromanage their kids' play, so the kids never get any confidence in their own abilities, but, oh my, are those parents ever so over-involved!