Pat chicken pieces dry with paper towels. Sprinkle chicken pieces with 1 teaspoon salt and 1/2 teaspoon pepper. In a shallow bowl, spread 3/4 cup all-purpose flour. Dredge chicken pieces, one at a time, in flour mixture, shaking off any excess.
Heat vegetable oil in a Dutch oven over medium-high heat. Add half the chicken pieces, skin side down, cooking 4-6 minutes until nicely browned. Remove browned chicken to a plate. Repeat with remaining chicken pieces, adjusting heat as need, so not to burn the flour.
Pour off all but 3 tablespoons of the oil. Reduce the heat to medium-low and add the onions, celery, garlic, and carrots and cook until veggies are soft, 7-8 minutes. Sprinkle remaining 3 tablespoons of all-purpose flour over veggies and stir to combine. Continue to cook for 1 minute. Mix in broth, poultry seasoning, cayenne pepper, 2 teaspoon salt, 1/4 teaspoon black pepper. Bring to a boil and reduce heat to low. Nestle chicken pieces into sauce along with any accumulated juices. Partially cover and simmer chicken 30-40 minutes, or until cooked through and tender.
Transfer chicken to a deep platter. Adjust gravy seasoning if needed, then pour over chicken. Sprinkle with parsley and serve.