Wow....that sounds like a nightmare all around.
I think IF you go again (IF !) then go at a different time.
We also have a very busy big park and I notice that it gets crazy busy with parents and daycares between 9:45-11 a.m. (usually after snack and before lunch)
So...I try and be sneaky and take the kids there at 8 a.m. or 8:30 am for an hour and we usually have the park all to ourselves and the kids can run around and I don't have to try and do headcounts over 20 other kids.

Also...make a plan with your assistant. It sounds like you didn't have one. I would say "how about you watch child A and child B and I will go with the older kids Child C and D and E and then we can switch if you want after 20 minutes. Always remember that people outside your home are judging you when you are a daycare and out with the kids.

I tend to put my infant in the swing. LOL Less risk of her getting injured/knocked over or picking stuff off the ground and infants just LOVE the swings. I swear she could be there all day and not complain as long as I fed her.

Good luck and I guess try to think of something to address it with the parent of your dck that saw you at the park and explain that it was just a hectic day and that you are going to try to make sure you think of a better system next time so that the parent feels like you are taking the situation as a learning experience and they won't feel nervous next time you take their little one to the park