Heat oil in large saucepan or stockpot over medium high heat. Add onions, carrots, celery and turnips and sauté until softened, about 10 minutes. Add broth, tomatoes, water, thyme, salt and pepper. Bring to a boil. Reduce heat, cover partially, and simmer for 15 minutes.
Increase heat to high, and add green beans and cabbage. Return to a simmer, reduce heat, and cook until vegetables are tender, 10 minutes. Taste soup and add more salt and pepper, if desired. Stir in parsley.
Serve sprinkled with Parmesan cheese or homemade croutons.
If you prefer softer vegetables, add the green beans and cabbage with the broth. Simmer 30 minutes or more, or until vegetables are desired consistency.