You will have no problem taking a soft sided cooler into the parks with juice and little snacks (no fermented juice for Mom and Dad though ;-)). As long as the cooler does not have wheels and there is no glass.
Technically, outside food and drinks aren't permitted but we look past it because we know that it is such a major expense to visit us. PLUS, most kids are incredibly picky eaters and drinkers and it may be an all day task to find something to apease them.