In a 2 1/2- to 3-quart pan over high heat, bring 1 1/2 quarts water to a boil. Rinse and drain wild rice and add to water. Return to a boil, cover, reduce heat, and simmer until rice begins to split, 40 to 45 minutes. Drain.
Meanwhile, put nuts in a 10- by 15-inch baking pan. Bake in a 325° (if roasting turkey at that temperature) or 350° (if using a different oven than one for turkey) regular or convection oven until lightly browned, 15 to 20 minutes.
In a 5- to 6-quart pan over medium-high heat, stir celery and onion in butter until onion is limp, 7 to 10 minutes. Add white rice and stir until edges turn opaque, 3 to 4 minutes. Add broth, apricot nectar, brandy, thyme, sage, nutmeg, and pepper. Bring to a boil, cover, reduce heat, and simmer until rice is tender to bite, 15 to 20 minutes. Stir in bread-cube stuffing, apricots, wild rice, and toasted nuts. Add salt and more pepper to taste. Cover and cook over low heat, stirring occasionally, until bread cubes are evenly moist, 7 to 12 minutes. Pour into a 3-quart casserole and cover tightly.
Bake in a 325º (if roasting turkey at that temperature) or 350º (if using a different oven than one for turkey) regular or convection oven until dressing is hot in the center, 30 to 40 minutes. For a crustier top, uncover casserole for the last 15 to 20 minutes. Sprinkle with parsley, if desired.