You are allowed to bring in "small, soft-sided, coolers" into any of the parks. The prohibitions are nothing in glass and no alcoholic beverages.
At the water parks if you can carry, drag, or wheel the cooler it is allowed. Again, nothing in glass and no alcoholic beverages. The biggest cooler I have ever seen anyone bring into TL was about four feet long, two feet high and two feet wide with a handle and wheels. Everything it it was legal except the jar of pickles.