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Preheat oven to 375 degrees.
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Heat 2 tablespoons olive oil in a large skillet over medium heat. Season chicken with salt and pepper; working in batches, add chicken to skillet, skin-side down, and cook until golden brown, 5 to 7 minutes. Turn chicken and cook 3 minutes more. Transfer to a large baking dish.
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Drain oil from pan and return to medium heat. Add white wine to deglaze and use a wooden spoon to scrape up browned bits from bottom of pan. Whisk in red-wine vinegar, mustard, light-brown sugar, and dates; simmer until reduced by half, 3 to 4 minutes.
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In a medium bowl, mix together remaining tablespoon olive oil, thyme, olives, capers, and cherry tomatoes; season with salt and pepper and toss to combine. Evenly distribute olive mixture over chicken and top with pan sauce. Transfer to oven and bake until chicken is cooked through, 55 to 65 minutes. Sprinkle with parsley and serve.
