We took a small, soft-sided cooler into the parks everyday. Our boys were 3.5 and 17 months during our trip last April. I didn't want to mess around with sippy cups, so I packed juice boxes and the small bottles of water for them to drink from. It was easier than worrying about losing cups.
At that age, my boys wouldn't finish a whole bottle of water, for instance, so it was nice to put it back in the cooler for later. We had our double stroller with us, so it wasn't a big deal to put it in the basket. We have a car and drive to the parks, so we also don't have to fold the stroller up on a bus, while holding 2 kids and all our other stuff.
Security had us open the cooler at the gates, but it wasn't a big deal, and noone ever questioned it. I say bring one if you have the means to carry it with you all day. I also brought plenty of snacks into the parks for them, too.