Preheat the oven to 425°. In a large, deep ovenproof skillet, melt the butter in the oil. Season the chicken with salt and pepper and dust with the flour, tapping off the excess. Add the chicken to the skillet skin side down and cook over high heat, turning once, until browned, 6 minutes. Transfer to a plate.
Add the shallots, parsnips and rosemary to the skillet and cook, stirring, for 1 minute. Add the wine and boil until reduced by half, about 3 minutes. Add the broth and bring to a boil. Nestle the chicken skin side up in the skillet, tucking it between the parsnips. Transfer the skillet to the middle rack of the oven and braise the chicken uncovered for about 25 minutes, until cooked through.
Turn the broiler on. Broil the chicken for 3 minutes, until the skin is crisp. Return the skillet to high heat and boil until the sauce is thickened, 3 minutes. Discard the rosemary sprig. Transfer the chicken and vegetables to bowls, garnish with parsley; serve.