Whisk together juice, oil, vinegar, garlic, oregano, and salt and pepper. Cover and refrigerate until ready to use. Whisk or shake vigorously before using.
CHICKEN: Preheat broiler at least 15 minutes prior to using. Broil chicken halves, turning once, for about 30 minutes or until skin is golden-brown and juices run clear when bird is pierced with a fork. Remove chicken from broiler, leaving broiler on. Using a very sharp knife, cut each 1/2 into about 6 pieces (leg, thigh, wing, 3 small breast pieces). Place chicken on a baking sheet (that will fit in the broiler) with sides. Pour lemon sauce over the chicken and toss to coat well. If necessary, divide sauce in 1/2 and do this in 2 batches. Return chicken to broiler and broil for 3 minutes. Turn each piece and broil for an additional minute. Remove from broiler and portion the chicken onto each of 6 warm serving plates. Pour sauce into a heavy saucepan. Stir in parsley and place over high heat for 1 minute. Pour an equal amount of sauce over each chicken and serve with lots of crusty bread to absorb the sauce.