In a large Dutch oven, combine the chicken, carrots, celery, onion, and chicken broth. Bring to a boil over medium-high heat.
Reduce the heat to low, cover, and simmer for 30-40 minutes until the chicken is tender and cooked through.
Remove the chicken from the pot and let it cool slightly. Shred the meat using two forks and discard any bones. Return the shredded chicken to the pot.
Stir the heavy cream and chopped parsley into the bubbling broth.
To make the dumplings, whisk together the flour, baking powder, and salt in a medium bowl. Gently stir in the buttermilk and melted butter until a soft dough forms.
Drop spoonfuls of the dough into the simmering broth. Cover the pot with a tight-fitting lid and simmer undisturbed for 15 minutes, or until the dumplings are cooked through and fluffy. Serve hot.