Combine hen, water, onion, and celery in a large stock pot; bring to a boil. Reduce heat; simmer, uncovered, 3 hours.
Remove hen from broth; bone and chop meat. Set white meat aside; reserve dark meat for use in other recipes. Strain broth, and discard vegetables. (Chicken and broth may be refrigerated overnight.) Skim off and discard excess fat from broth.
Return broth to stock pot; bring to a boil. Stir in rice, parsley, salt, and pepper. Reduce heat; cover and cook 20 minutes or until rice is tender.
Gradually add milk to flour, stirring constantly, until smooth. Stir flour mixture and white meat into rice mixture. Cook over medium heat, stirring frequently, until thickened and thoroughly heated. Serve immediately in warm soup bowls.