Stir together egg, beef, onion, bread crumbs, 1 tablespoon of the parsley, the Parmesan and 1/4 teaspoon each of the salt and pepper. Form into 1-inch meatballs (about 20) and set aside.
Stir together 1/4 cup ketchup, 1/4 cup broth and the vinegar. Place carrots, parsnips and onion in slow cooker and put meatballs on top. Drizzle with ketchup mixture. Cook on HIGH for 3-1/2 hours or LOW for 5 hours.
Stir together remaining tablespoon parsley, 1/4 teaspoon each salt and pepper, 1/4 cup ketchup, 1 tablespoon broth and cornstarch. Gently stir into slow cooker. Cook until thick, 5 minutes.