Cut breasts from duck and discard skin and all visible fat. Reserve legs and carcass for another use. Brown duck neck (halved if necessary) in a dry 2- to 3-quart heavy saucepan over moderate heat, turning once, about 10 minutes total. Transfer to a plate. Pat duck breasts dry and season with salt and pepper, then add to pan and brown over moderate heat, turning once, about 8 minutes total. Transfer to plate with duck neck. Add Armagnac to pan and deglaze by boiling, scraping up any brown bits, until most of liquid is evaporated. Add 2 cups water, shallots, garlic, sliced carrot, thyme, bay leaf, kosher salt, peppercorns, and duck breasts and neck with any juices on plate and cook at a bare simmer, covered, 1 hour. Discard neck, then transfer breasts to a plate and cool. Shred duck meat, then chop. Pour broth through a sieve into a bowl, discarding solids. (If liquid measures less than 2 cups, add water.) Return to cleaned pan and add chopped carrot. Simmer, covered, until carrot is tender, 3 to 5 minutes. While carrot is cooking, sprinkle gelatin over remaining 2 tablespoons water in a cup and let soften 1 minute. Stir softened gelatin into hot broth until dissolved, then stir in duck meat, parsley, and salt and pepper to taste. Ladle into crocks (you`ll have about 3 cups total) and chill, covered, until set, 3 to 4 hours. Let stand at room temperature 20 minutes before serving.