Adjust oven rack to middle position and heat oven to 400 degrees. Combine giardiniera and brine, 1/4 cup oil (4 tablespoons), parsley and pepper in bowl. Set aside.
Pat chicken dry with paper towels and season with salt and pepper. Heat remaining 1 tablespoon oil in 12-inch skillet over medium-high heat until just smoking. Add chicken, skin side down, and cook until well browned, 5 to 7 minutes. Flip chicken and transfer skillet to oven. Cook until chicken registers 175 degrees, about 15 minutes. Transfer chicken to platter and top with relish.