In a large saucepan, over medium heat, add the olive oil. Season the duck pieces with salt and pepper. When the oil is hot, sear the duck, fat side down for about 4 to 6 minutes. Turn over the duck and continue to sear for 4 minutes. Remove the duck from the pan and set aside. Add the onions and celery to the pan. Season the vegetables with salt and cayenne. Saute the vegetables until wilted, about 4 minutes. Add the sausage and continue to saute for 4 minutes. Stir in the beans, bay leaves, and garlic. Add the duck stock, water and fresh thyme. Place the duck pieces back into the pan. Bring the liquid to a boil and skim off any cloudy scum that has risen to the service. Reduce the heat to a simmer and cook, uncovered for 3 hours, stirring occasionally or until the meat and beans are tender. Reseason if necessary and stir in the green onions and parsley.