In a large saucepan cook carrots and parsnips in a small amount of water, covered, for 3 minutes. Drain.
Transfer the partially cooked carrots and parsnips to a 13x9x2-inch baking pan. Sprinkle with parsley, desired herb, and salt. Drizzle with oil. Toss gently to coat vegetables. Cover the pan with foil.
Bake in a 375 degree F oven for 30 minutes, stirring vegetables once. Stir in squash pieces. Cover and bake about 20 minutes more or just until vegetables are barely done. Remove vegetables from oven.
Increase oven temperature to 450 degree F. Stir the brown sugar into vegetables until thoroughly combined. Return vegetables to oven and bake, uncovered, for 15 to 20 minutes more or until vegetables are tender and glazed. Transfer to a serving dish. If desired, garnish with fresh thyme. Makes 8 servings.