Preheat oven to 350°F; coat a 9-by-13-inch baking dish with cooking spray.
In a large skillet over medium-high heat, warm oil and butter together until butter melts. Add onion and celery and cook, stirring often, until softened but not browned, about 5 minutes. Add turkey sausage and parsley and cook about 2 minutes longer, breaking up sausage with back of a spoon.
Place corn bread in skillet and stir well to combine. Add chicken broth, then season mixture with salt and pepper and transfer to prepared baking dish. Cover tightly with foil and bake until stuffing is warmed through, about 20 minutes. Remove foil and bake 15 minutes longer to crisp top.