Preheat oven to 400°F. In a greased 2-quart rectangular baking dish, layer half of the rutabaga, thyme, salt, pepper and Gruyere; repeat layers. Combine cream and chicken broth; pour over the top. Cover with foil and bake 1 hour.
Remove foil. Top with Parmesan and bake 10 minutes more or until rutabaga is tender and cheese is browned. Let stand 10 minutes before serving.