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In a Dutch oven, heat oil over medium heat and cook carrots, celery, onion, and garlic for 5 minutes.
Add 1½ cups water, broth, bay leaves, thyme, salt, and pepper and bring to a boil.
Add chicken, reduce heat, and simmer for 20 minutes. Add potatoes and simmer until cooked.
Whisk flour into ½ cup water and mix into mixture. Remove bay leaves.
Whisk together flour, cornmeal, baking powder, baking soda, and salt.
Cut in butter until coarse crumbs form. Stir in buttermilk and 2 tablespoons parsley, making a sticky, spoonable dough.
Drop by 8 spoonfuls into simmering stew.
Cook, covered without lifting the lid, until dumplings are no longer doughy underneath, for about 15 minutes.