Place the chicken thighs, onion, carrots, and celery in a large pot. Add the chicken broth, thyme, rosemary, garlic, and bay leaf. Season with salt and black pepper.
Bring the mixture to a boil over medium-high heat, then reduce to a simmer. Cook for 15 minutes until the chicken is cooked through and the vegetables begin to soften.
While the broth simmers, stir together the flour, baking powder, and salt in a bowl. Pour in the milk and melted butter and mix until a soft dough forms.
Drop spoonfuls of the dough directly into the simmering broth. Cover the pot and cook for 12 minutes without lifting the lid so the dumplings steam properly.
Remove the bay leaf. Stir in the fresh parsley and adjust salt and pepper if needed before serving.