My husband was a SAINT and carried a backpack, a backpack cooler, and sometimes a child. We did stow on bag in our stroller most of the time. I also carried a small cross-body purse.
In my purse:
A small wallet with cash, credit cards, and drivers licenses.
Meal confirmation papers.
Autograph book and Sharpie. (Keep this handy! You may need it quickly!)
Map and schedule.
In our backpack:
External battery for cell phone. (Mine ran down QUICKLY in the parks.)
Bandaids and first aid.
A change of clothes for each child.
A hand towel. (I used this for diaper changes and for various other things. Came in handy!)
Squeeze applesauces. (My kids LOVE these.)
Bottles of water.
Squirt drink flavors.
Bottles of apple juice for boys.
What are your necessities? Anything I missed?