In a 6- to 8-quart Dutch oven combine chicken, water, onion, salt, pepper, and bay leaf. Bring to boiling; reduce heat. Simmer, covered, about 1-1/2 hours or until chicken is very tender.
Remove chicken from broth. When cool enough to handle, remove meat from bones. Discard bones and skin. Cut meat into bite-size pieces; set aside. Discard bay leaf. Skim fat from broth.
Bring broth to boiling. Stir in carrot and celery. Simmer, covered, about 5 minutes. Stir in noodles. Simmer, covered, 5 minutes more or until noodles are tender but still firm. Stir in chicken and parsley; heat through.