Place the whole chicken in a large pot with water, celery, carrots, onion, salt, and pepper. Bring to a boil, then reduce heat and simmer for 60-75 minutes until chicken is fully cooked and tender.
Remove chicken from pot and let cool slightly. Shred the meat, discarding skin and bones, and set aside. Strain the broth and return it to the pot.
In a large bowl, whisk together flour, baking powder, and salt. Cut in the cold butter using a fork or pastry cutter until mixture resembles coarse crumbs. Stir in milk until just combined.
Bring the broth back to a gentle boil. Drop dumpling dough by spoonfuls into the simmering broth, cover, and cook for 15 minutes without lifting the lid.
Gently stir in the shredded chicken and cook for an additional 5 minutes. Garnish with fresh parsley and serve hot.