Put 2 tablespoons of the melted butter in a large skillet or sauté pan. Place the pan over medium heat and add the chopped onion and celery. Cook, stirring, for 3 to 4 minutes, until the onion is translucent.
Add the soup, mayonnaise, and milk to the skillet and stir until well combined. Remove the pan from the heat.
Put the cornbread dressing crumbs and chicken in a large bowl. Add the soup mixture and enough chicken broth to moisten; taste and add salt and pepper, to taste. Stir in the parsley and transfer the mixture to a lightly buttered 2 1/2 to 3-quart baking dish. Drizzle with the remaining 2 tablespoons of melted butter.
Bake the casserole for 30 to 40 minutes, until hot and thoroughly cooked.