Position a rack in the center of the oven and preheat to 425. Heat the olive oil in a large ovenproof skillet over high heat. Add the shallots and parsnips and cook until golden, shaking the pan, 2 minutes.
Whisk the apricot preserves, mustard, ginger and cumin in a bowl. Season the chicken with salt and pepper; toss with the apricot glaze.
Scatter the dried fruit in the skillet. Place the chicken, skin-side up, on top. Add 1/4 cup water and bring to a boil. Cover and cook until heated through, 6 minutes. Uncover and transfer the skillet to the oven. Cook until the chicken and vegetables are tender and golden, 20 to 25 minutes.
Push the chicken to the side of the skillet, then stir the vinegar into the pan juices (add up to 2 tablespoons water if the sauce is too thick). Serve from the skillet.