In a large soup pot, cook ground beef, onion, and celery over medium-high heat until meat is browned and vegetables soften, breaking up meat as it cooks. Drain fat.
Stir in 1 can of the broth, both cans of tomatoes, frozen vegetables, steak sauce, Worcestershire sauce, salt, and pepper. Bring to a boil. Reduce heat, cover and simmer for 20 minutes.
In a medium bowl, whisk together remaining can of broth and the flour; stir into hot soup.
Cook over medium-high heat until thickened and bubbly. Cook and stir for 1 minute more.
Ladle soup into bowls; sprinkle each serving with parsley.
