Heat water to boiling; add salt, carrots, onion, celery and chicken. Cover and simmer (do not boil) 2½ to 3 hours, or until chicken is tender.
Remove chicken from broth and keep warm. Strain broth and measure; add enough water to make 3 cups.
Return broth to saucepan and heat. Put milk and flour in jar with cover and shake until mixture is smooth. Add slowly to hot broth, beating with wire whisk to keep smooth. Cook about 5 minutes. Add chicken to gravy.
Make Parsley Dumplings and add to simmering chicken about 20 minutes before serving.