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1. Heat the oil in a large pot or Dutch oven over medium-high heat. Season the chicken with 1/2 teaspoon each salt and pepper. Brown in batches, 4 to 6 minutes per side. Transfer to a plate; reserve the pot.
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2. Add the celery, carrots, onions, thyme, and garlic to the drippings in the pot and cook, stirring, until the vegetables begin to soften, 5 to 7 minutes. Add the chicken, bay leaves, and 10 cups water. Bring to a simmer and cook until the chicken is cooked through, 25 to 30 minutes. Discard the bay leaves and transfer the chicken to a plate; let cool. Shred the chicken with 2 forks and return it to the pot (discarding the skin and bones).
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3. Whisk together 1/2 cup of the flour, 2 cups of the cooking liquid, and 1/4 teaspoon each salt and pepper in a small bowl. Slowly whisk the flour mixture back into the pot and simmer until slightly thickened, 8 to 10 minutes.
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4. Make the dumplings: Whisk together the remaining 2 cups of flour, the baking powder, baking soda, and 1/4 teaspoon each salt and pepper in a medium bowl. Whisk in the butter, buttermilk, and parsley. Reduce heat to low and drop the mixture into the broth in 8 large spoonfuls. Simmer, covered, until the dumplings are firm, 12 to 15 minutes. Serve sprinkled with parsley.
