Cook rice according to package directions; set aside.
Combine water and next 3 ingredients in a Dutch oven; bring to a boil. Cover, reduce heat, and simmer 20 minutes; add chicken livers, and cook 10 to 12 minutes or until tender. Drain, reserving 3/4 cup liquid. Remove meat from wings; coarsely chop wing meat, gizzards, hearts, and livers. Set meat aside.
Brown sausage in Dutch oven, stirring until it crumbles; drain and set aside.
Melt butter in Dutch oven; add onion, green pepper, and celery. Cook over medium heat 5 minutes or until tender, stirring often. Add sausage, chopped meat, and reserved liquid. Bring to a boil; cover, reduce heat, and simmer 15 minutes. Stir in cooked rice and parsley. Serve warm.