Sprinkle the chicken on both sides with the Cajun seasoning. Place the chicken in a baking dish and drizzle with the vinegar and oil. Shake the pan gently to coat the chicken underneath. Cover and refrigerate for at least 1 hour or overnight for the best flavor.
When you are ready to grill, move the chicken to the counter to take off the chill. Preheat the grill to medium-high heat, about 400 degrees F.
Lightly oil the grill and place the chicken on it, smooth-side down. Cook until grill marks appear and the chicken lifts up easily, about 4 minutes. Rotate the chicken by a 45-degree angle and cook to mark, about 3 minutes. Flip the chicken to the other side and cook until cooked through, about 2 minutes. Top with the parsley. Serve hot.
To oil the grill, fold a half sheet of paper towel a few times until it is a small square. Dip one side in oil and, using grill tongs, rub the oiled paper towel on the grill in the areas that you will cook the chicken.