Heat oil in a large stockpot over medium-high heat. Add carrots, celery, onion, and parsley. Add salt and pepper and cook vegetables until they sweat, about 5 minutes. Drop the chicken right into the stockpot and add water to cover by 3 to 4 inches. Bring to a boil and cook until no longer pink in the center and juices run clear, about 1 hour. Meanwhile, place flour in a large bowl; cut in shortening. Add buttermilk slowly until flour comes together in a a large ball. Knead dough slightly; do not overdo it or dumplings will not be fluffy. Roll dough out and cut into dumpling strips. Pull cooked chicken out of pot and let cool until safe to handle, about 15 minutes. Reserve stock. Shred chicken meat. Pull all veggie chunks out of stock. Add shredded chicken back to the pot with stock and bring to a boil. Add dumplings slowly, 1 at a time, stirring constantly to keep from sticking. Cook until stock is thickened and dumplings are no longer cream-colored, but almost brownish-grey, 5 to 10 minutes.