In the bowl of a food processor combine the garlic, onion, parsley, salt, pepper, rum, sugar and lime juice. Process until finely chopped and drizzle in olive oil.
Generously rub the pork with the marinade and marinate overnight of for several hours. Reserve any remaining marinade for later.
Peel the chayote and slice into 1/2-inch thick pieces. Brush with any remaining marinade. Preheat a heavy roasting pan. Add several tablespoons of vegetable oil to the pan and place the roast, with the chayote into the pan. Roast for 1 1/2 hours or until the internal temperature reaches 155 degrees. Brush with any remaining marinade while the roast is cooking. Remove from the oven and rest for 20 minutes before carving. Serve with the chayote and green beans.