Combine chicken, water, celery, carrots, onion, salt, and pepper in large pot. Boil; reduce heat and simmer 60-75 minutes until chicken is fully cooked and tender.
Remove chicken; cool slightly, shred, discard skin and bones, and set aside. Strain broth; return to pot.
Whisk flour, baking powder, and salt in large bowl. Cut in butter with fork or pastry cutter to coarse crumbs; stir in milk just until combined.
Bring broth to gentle boil; drop in spoonfuls of dough. Cover; cook 15 minutes without lifting lid.
Gently stir in chicken; cook 5 minutes. Garnish with parsley; serve hot.