Loosen the skin on the breast and legs. Rub the flesh of the chicken, under the skin, with the lime zest. Mix the curry powder, parsley, salt and pepper together and rub all over the chicken. Place in a resealable plastic bag. Let the chicken sit overnight in the fridge.
Preheat the grill to medium. Brush the grill with vegetable oil.
Place the chicken on the grill skin-side down and grill until the skin becomes golden brown, about 15 minutes. Flip the chicken over, close the cover and continue grilling until just cooked through, about 25 minutes. Remove from the grill, loosely tent with foil and let rest before serving.