Season chicken with 1 tsp salt and pepper. Brown skin-side down in oil in a Dutch oven over medium-high heat 6 minutes. Turn; cook 2 minutes more.
Transfer chicken to a plate; pour off all but 1 tbsp fat. Cook onion, celery, and carrots 5 minutes until onion softens.
Return chicken with broth and bay leaf. Bring to a gentle simmer; cook partially covered 30 to 35 minutes until chicken reaches 175°F.
Remove chicken; discard skin and bones. Pull meat into bite-size pieces; discard bay leaf.
Whisk 1/4 cup flour with 1/2 cup pot broth until smooth. Stir into pot; simmer 3 minutes until lightly thickened.
Whisk remaining 1 1/2 cups flour, baking powder, and remaining 1/2 tsp salt. Rub in butter until pea-size pieces remain; stir in buttermilk.
Return chicken to simmering broth. Drop 12 heaping dough spoonfuls over surface, leaving gaps.
Cover tightly; cook 10 minutes without lifting lid. Uncover; simmer 5 to 7 minutes until dumpling interiors are dry, not gummy. Finish with parsley.