HAHAHAHAHAHAAA!!! Oh boy did I need that laugh today.
I am assuming you are kidding...Buuuuuuuuuttt, in case you aren't.....No you may not bring a WAGON into the park to pull your child and cooler around in. You can get a stroller (or as the other poster mentioned, a double stroller) and put your SMALL SOFT SIDED COOLER in the unused half of the stroller.
I can just picture somebody hauling a radio-flyer through the park with a child in back with an igloo cooler. Hopefully that isn't what you had in mind.