Preheat the oven to 400F. Whisk together the oil, vinegar, salt, red pepper flakes, oregano, and garlic in a 9 x 13-inch baking dish. Add the chicken to the marinade in the dish, turning each piece to coat. Cover with plastic wrap, and refrigerate at least 1 hour or up to 8 hours.
Stir together the bread crumbs, cheeses, parsley, and pepper in a medium bowl. Line a rimmed baking sheet with parchment paper. Oil the parchment.
Dredge each chicken piece in the bread crumb mixture to coat; shake off excess. Transfer to the prepared baking sheet.
Bake the chicken until the bread crumbs are golden and the chicken is cooked through, 45 to 50 minutes. Using a spatula, loosen the chicken from the sheet before serving.