Start by taking the fish out of the refrigerator while you prep the ingredients and preheat the broiler.
Before preheating the broiler, be sure to get the top oven rack into a position so the fish is about 3 to 4 inches from the broiler heat source. You don't want to be messing with the rack after it gets hot.
In a medium sized bowl, mix the mayonnaise and mustard and stir in the parsley.
Place the flounder fillets onto a baking sheet. I sprayed my baking sheet with a little oil first to prevent sticking.
Brush the mayo/mustard mix evenly over the fillet(s) and place the pan under the broiler about 3 to 4 inches from the heat source.
The recipe says to "broil for about 1 minute or until golden brown on top and the fish is just cooked through." Now maybe Craig Claiborne's flounder fillets were thinner than mine, but I found it took 2 to 3 minutes for them to be cooked through so keep a close watch on them when cooking.
Serve with potatoes or rice, a favorite vegetable and don't forget to add the lemon wedges.