Heat 1 1/2 teaspoons oil in a large Dutch oven over medium-high heat. Add half of beef, pork, and chicken; sauté 5 minutes or until browned. Remove beef mixture from pan; repeat procedure with remaining 1 1/2 teaspoons oil, beef, pork, and chicken. Add 3 cups water, salt, 1/4 teaspoon black pepper, and broth, scraping pan to loosen browned bits. Add beef mixture, onion, bell pepper, carrot, and garlic to pan; bring to a boil. Reduce heat, and simmer, uncovered, 1 hour or until beef is tender, stirring occasionally.
Add remaining 1/2 cup water, potato, beans, corn, okra, thyme, and 1 tablespoon parsley; bring to a boil. Reduce heat, and simmer 30 minutes or until potato is tender. Stir in remaining 1/4 teaspoon black pepper, sugar, and the next 4 ingredients (through tomatoes); simmer 30 minutes or until mixture is thick. Sprinkle with additional parsley, if desired.