Heat the oil in a medium saute pan over medium heat. Add the ginger and garlic; cook until soft, about 2 minutes. Add the mustard seeds; cook 30 seconds. Add 1/2 cup water and the lime zest and season with salt and pepper. Bring to a boil and cook 1 minute. Transfer the mixture to a blender; blend to a smooth paste, adding more water if needed. Let cool.
Rub the ginger paste over the entire chicken, including underneath the skin. Cover the chicken and marinate in the refrigerator, at least 2 hours and up to 8 hours. Remove the chicken from the refrigerator about 30 minutes before cooking.
Combine the cardamom, cinnamon and cloves in a bowl. Cover with cold water and soak at least 1 hour and up to 4 hours.
Combine the red onion and 1 tablespoon salt in a bowl and let sit at room temperature 30 minutes. Rinse with cold water and drain well. Put the onion in a bowl and add the parsley and sumac.
Heat your grill to high for indirect grilling.
Drain the spices and add them evenly over the hot coals in a charcoal grill or to the smoker box of a gas grill. Put the lid on the grill and allow smoke to fill the grill, about 10 minutes.
Season the chicken on both sides with salt and pepper. Put the chicken, skin-side down, on the cooler side of the grill (indirect heat), close the cover and grill until the skin is lightly golden brown with faint grill marks, 22 to 25 minutes. Turn the chicken skin-side up, move to the hotter side of the grill (direct heat) and continue grilling until the skin is golden brown, 15 to 20 minutes. Flip the chicken once more and continue grilling over direct heat until the skin is golden brown and crisp and an instant-read thermometer inserted into the thigh registers 165 degrees F, about 8 minutes.
Remove the chicken to a cutting board, tent loosely with foil and let rest about 10 minutes before carving. Top the chicken with some of the onion-parsley relish.