Add olive oil, parmesan, garlic, oregano, thyme, salt, and pepper to blender. Blend until ingredients are fully combined, pausing to scrape down sides of blender as needed. When mixture is combined and develops pasty consistency, set blender aside.
Preheat oven to 425° Fahrenheit. Place chicken pieces on cutting board and pat chicken completely dry on all sides with paper towels. Season chicken pieces on all sides with salt and pepper to taste, then set chicken aside.
Heat large cast-iron skillet over medium-high heat. When skillet is warm, add olive oil and continue heating, swirling skillet occasionally to fully coat pan in oil, until oil is hot and shimmering.
When oil is hot, place chicken pieces in skillet skin-side down. Cook, undisturbed, 7 to 8 minutes or until skin is brown and releases easily from skillet. Note: If skin sticks to skillet, allow to cook another 1 to 2 minutes undisturbed. Check again and repeat as needed until skin releases easily. Be careful not to burn chicken.
When skin of chicken is browned and releases easily from skillet, carefully flip chicken over so skin-side faces up. Evenly spread or brush parmesan-herb paste over skin of chicken, using entire batch of paste to cover chicken well.
Immediately place cast-iron skillet in preheated oven. Roast chicken pieces 18 minutes, then begin checking internal temperature of each piece for doneness. Chicken pieces are ready when internal temperature reaches 160° Fahrenheit. Note: Exact cooking time will vary based on size and thickness of chicken pieces.
As chicken pieces reach desired temperature, transfer pieces from skillet to plate and cover with foil to keep warm. Continue roasting chicken and checking temperatures until all chicken pieces have reached 160° Fahrenheit and have been transferred to plate. Set chicken aside, still covered with foil.
Discard any liquid from cast-iron skillet and return skillet to medium-high heat. Add red wine vinegar to skillet and deglaze pan, scraping up any browned bits of chicken or parmesan-herb paste that may be stuck to bottom of skillet. Simmer vinegar over medium-high heat 4 to 5 minutes or until liquid has reduced slightly.
Transfer chicken pieces to serving plates or serving platter. Spoon red wine vinegar reduction over chicken, garnish with parsley if desired, and serve chicken warm with preferred sides.