Preheat oven to 325°F. Sprinkle both sides of brisket with 1 tablespoon of the salt and 1 1/2 teaspoons of the pepper. Heat oil in a Dutch oven over medium-high. Add brisket, and cook until deeply browned, 5 to 6 minutes on each side. Transfer brisket to a plate.
Stir together balsamic vinegar ketchup, cola, chicken broth, onion soup mix, and herbes de Provence in a large bowl. Pour slowly into Dutch oven, and stir and scrape to loosen browned bits from bottom of Dutch oven. Return brisket and any juices to Dutch oven. Bring to a boil over medium-high, and cover with foil followed by a tight-fitting lid. Transfer Dutch oven to preheated oven.
Bake 2 hours. Turn brisket over. Tuck in carrots and parsnips, and sprinkle onion into liquid around edge of brisket. Replace foil and lid, and continue baking until brisket is fork tender, about 1 hour.
Use a slotted spoon to transfer brisket, carrots, and parsnips to a serving platter. Cover with foil to keep warm. Puree cooking liquid in Dutch oven with an immersion blender to make gravy. Simmer gravy over medium until heated through. Stir in bourbon, and sprinkle with remaining 1/2 teaspoon salt and 1/4 teaspoon pepper.
Slice brisket. Spoon some of the hot gravy over brisket and vegetables, and sprinkle with thyme. Serve remaining gravy in serving dish on table.