You're not supposed to but you can "smuggle it in" no one really cares but they will if you have a cooler
Yes, security will check any bags you have, and they definately discourage food or drinks. However, they wont give you a hard time if you do tell them you are on a special diet, aka gluten free, or dairy free, etc.
Half the people go there with kids (we did), and they didnt care what we brought into the park, which was a lot of drinks and food for our picky eaters. It did help that we had a stroller!