Mash the garlic, salt, butter and hot pepper sauce together in a bowl to make a paste. Spread under and over the skin of the chicken pieces.
Preheat gas grill on HIGH, then reduce heat to LOW and brush the grids with vegetable or olive oil to prevent sticking. Place chicken pieces on the grill and cook, turning occasionally, for 30 – 45 minutes.
Meanwhile, in a large, deep bowl, whisk together the lemon and orange juices, orange zest, olive oil, citrus liqueur and barbecue sauce. Stir in olives.
Add the hot, cooked chicken pieces to the bowl and turn them in the dressing. Serve the chicken, its dressing and the olives garnished with parsley and lemon halves.