Place a skillet or grill pan over medium heat and add the olive oil.
In a bowl, combine the chicken, mozzarella, cheddar, cream cheese, garlic, parsley, and a pinch of salt and pepper. Stir until the mixture is uniform and creamy.
Divide the filling among the tortillas into four equal portions. Roll each tightly into a compact wrap.
Put the wraps in the heated skillet with the seam facing down. Cook for 2–3 minutes on each side, until they are golden and the cheese inside is fully melted.
Cut each wrap in half and serve right away. If desired, sprinkle additional parsley on top for a fresh garnish.