Pre-heat oven to 450°F.
Place chicken on a rimmed baking sheet and toss with turmeric and 2 tablespoons olive oil; season with salt and pepper.
Make sure chicken is skin-side up, then pour vinegar over and around chicken and place in the pre-heated oven.
Bake chicken, without flipping, until cooked through and deeply browned all over; 25 to 30 minutes.
Meanwhile, in a small bowl, combine olives, garlic, parsley, the remaining 4 tablespoons olive oil and 2 tablespoons water; season with salt and pepper.
Once chicken is cooked, remove baking sheet from the oven and transfer chicken to a large serving platter, leaving behind any of the juices and bits stuck to the pan.
Make sure the baking sheet is on a sturdy surface, then pour the olive mixture onto the sheet.
Using a spatula or wooden spoon, gently scrape up all the bits of chicken left behind, letting the olive mixture mingle with the rendered fat and get increasingly saucy.
Pour olive mixture over the chicken and serve.