We'll usually put sunscreen on before we leave the hotel room, time permitting. But we'll take a can of spray on sunscreen to the park just in case we are going to be there for a while and need to reapply. I just carry the bottle in my cargo shorts, which I always wear to the parks for the purpose of not having to carry a bag or go through bag check.
You could also bring a small bottle, put it in a ziplock bag, and stash it in your pocket so it doesn't accidentally come open and make a mess.