Always wear closed-toed foot gear in the parks, at least in my opinion. It's too crowded, too full of strollers, wheelchairs and people not playing attention for you not to get stepped on or rolled over at some point (People stomped on my feet three times in three days last weekend alone).
I am always very comfortable in Gore-Tex hiking shoes, which are waterproof but let moisture escape. They also have excellent ankle support, which, considering you are sometimes on your feet for 10 miles or more a day, is very helpful.