Heat oil in a heavy 4-qt. pan over medium heat; add onion, garlic, and celery.
Cook, stirring, until onion is limp.
Stir in tomatoes (break up with a spoon) and their liquid, broth, Italian herbs, and rice.
Cover and simmer for 15 minutes.
Add green beans and simmer, covered, for about 10 minutes longer or until rice and beans are tender to bite.
Meanwhile, remove skin and bones from fish and cut into about 1-in. pieces. Add to soup and simmer just until fish flakes readily when prodded with a fork (about 5 minutes).
Season with salt and pepper to taste.
Sprinkle individual servings with parsley and pass Parmesan cheese to spoon over each serving, if you wish.