Place chicken, celery, onion, parsley, salt, pepper and water in Dutch oven. Cover and heat to boiling; reduce heat to low. Cook over low heat about 2 hours.
Remove chicken and vegetables from Dutch oven. Skim 1/2 cup fat from broth; reserve. Remove broth; reserve 4 cups and refrigerate the rest for another use.
Heat reserved fat in Dutch oven over low heat. Stir in 1/2 cup Bisquick. Cook, stirring constantly, until mixture is smooth and bubbly; remove from heat. Stir in reserved broth and heat to boiling, stirring constantly. Boil and stir about 1 minute.
Return chicken and vegetables to pot; cook until heated through. Mix remaining 2 cups Bisquick and the milk until soft dough forms. Drop dough by spoonfuls onto hot chicken mixture (do not drop directly into liquid). Cook uncovered over low heat 10 minutes. Cover and cook 10 minutes longer.