Arrange the turkey parts on a large rimmed baking sheet. Let come to room temperature for 1 hour.
Meanwhile, using a food processor, pulse the garlic until chopped bits stick to the bowl. Add the parsley and coarsely chop. Add the rosemary, thyme, 1/4 cup EVOO, 1 tbsp. salt and 1 tsp. pepper and pulse until finely chopped but not smooth. Using your hands, slather the herb paste on the turkey parts.
Preheat a gas grill to medium (350 degrees ). Brush the hot grill grate clean, then grease it with a bunched-up paper towel dipped in EVOO. Add the turkey parts skin side up, cover the grill and cook for 30 minutes, adjusting the heat as necessary to maintain a temperature of 350 degrees to 375 degrees . Flip the turkey parts and cook for 20 minutes, then flip again and cook until the meat registers 150 degrees to 155 degrees on an instant-read thermometer, 10 to 20 minutes longer (breasts will take about 10 minutes longer than thighs and drumsticks).
Transfer the turkey parts to a large carving board, tent with foil and let rest for at least 15 minutes before carving, while you make the gravy.