Adjust oven rack to the middle position and preheat oven to 300 degrees. In a Dutch oven, combine beef brisket, chicken broth, water, chopped carrots, celery, onion, bay leaves, peppercorns, thyme and allspice berries. Cover and bake until a fork inserted into brisket goes in and out easily, about 4 1/2 to 5 hours.
Transfer meat to a 9-inch by 13-inch baking dish. Strain liquid through a fine-mesh strainer into a large bowl and discard the solids. Skim fat from liquid. Pour 1 cup of the cooking liquid over the beef, cover the dish tightly with aluminum foil, and allow to rest for 30 minutes.
Return the remaining cooking liquid to the Dutch oven, add the butter and bring to a simmer over medium-high heat. Add the potatoes and simmer until softened, about 10 minutes. Add the carrot halves and cabbage, cover, and cook until tender, about 10 to 15 minutes. Using a slotted spoon, transfer the vegetables to a serving platter, seasoning with salt and pepper to taste.
To serve, slice the brisket across the grain in 1/4-inch slices. Arrange slices on a platter. Using a slotted spoon, transfer the cooked vegetables to the platter and lightly moisten with cooking liquid as needed until served. Garnish with fresh parsley if desired.