Preheat the oven to 450 degrees F. Season the chicken with salt and pepper.
Heat the olive oil in a large ovenproof skillet over medium-high heat. Working in batches if necessary, add the chicken, skin-side down, and cook until golden, about 6 minutes. Flip and cook 2 to 3 more minutes, then transfer to a plate. Pour off all but 2 tablespoons of the drippings.
Add the onion and apples to the skillet and season with salt and pepper. Cook until slightly softened, about 4 minutes. Mix the broth with the mustard, then add to the skillet and bring to a boil. Arrange the chicken, skin-side up, in the skillet. Transfer to the oven and roast until the chicken is cooked through, 15 to 20 minutes.
Mix the butter and flour to form a paste. Use a slotted spoon to transfer the chicken, apples and onion to plates. Bring the pan juices to a simmer, whisk in about half of the butter-flour mixture and boil to thicken, 2 minutes. Continue to cook, adding more of the butter-flour mixture as needed to make a slightly thick gravy. Season with salt and pepper. Pour over the chicken and sprinkle with parsley.