Preheat the oven to 450˚. Generously season the chicken thighs on both sides with salt and pepper. Place skin-side down in a large ovenproof nonstick skillet and set over medium-high heat. Cook until the skin is very golden, about 10 minutes. Remove the chicken to a plate.
Meanwhile, peel the rutabaga and turnip, cut in half and thinly slice. Arrange about half of the vegetables in the drippings in the skillet, going around in concentric circles and overlapping slightly to make an even layer. Sprinkle with half each of the parmesan and rosemary and a big pinch of salt and pepper. Repeat with the remaining vegetables, parmesan and rosemary; season with salt and pepper. Pour the chicken broth over the vegetables and cook until the broth is bubbling, 2 to 3 minutes. Place the chicken skin-side up on top of the vegetables.
Transfer the skillet to the oven and roast until the vegetables are tender and the chicken is cooked through, 20 to 25 minutes. Meanwhile, stir the mustard and horseradish in a small bowl. Serve the chicken and vegetables with the mustard sauce on the side. Sprinkle with the parsley.