Serve with a fruit salad.
Combine gizzards, hearts, necks, and stock in Dutch oven and simmer, covered, 30 minutes. Add livers and cook 15 minutes longer. Drain. Cool stock, chill, and skim fat from top. Trim gristle from gizzards and chop with hearts and livers. Pick meat from necks. Return stock and meat to pot. Bring to a boil, add celery, onions, and parsley, and simmer, partially covered, 10 minutes. Add rice, reduce heat, cover, and simmer 20 minutes longer, or until rice is cooked Season with salt and pepper. Garnish with chopped parsley.