Dice the onion, chop the green and red bell peppers into ½-inch pieces, and mince the garlic. Open the diced tomatoes and tomato sauce (do not drain the tomatoes) and measure out the broth, Worcestershire sauce, brown sugar, and seasonings.
Heat the olive oil in a 6-quart Dutch oven over medium-high heat. Add the ground beef, break it into small crumbles, and cook for about 7 minutes until no pink remains. Spoon off all but about 1 tablespoon of the rendered fat.
Add the onion, green bell peppers, and red bell pepper and cook for about 5 minutes, stirring often, until the onion is translucent. Add the garlic and stir for 30 seconds until fragrant.
Pour in the undrained diced tomatoes, tomato sauce, and beef broth. Stir in the Worcestershire sauce, brown sugar, Italian seasoning, dried parsley, 1 teaspoon salt, and black pepper, then tuck in the bay leaf. Bring to a boil over high heat, scraping up browned bits.
Reduce the heat to medium-low, cover, and simmer for 30 minutes, stirring every 10 minutes. A piece of green pepper should pierce easily with a fork but still hold its shape.
Combine the rice, 2 cups water, and ¼ teaspoon salt in a medium saucepan. Bring to a boil, stir once, cover tightly, and cook on the lowest heat for 15 to 18 minutes until the water is absorbed. Rest covered for 5 minutes, then fluff with a fork.
Remove and discard the bay leaf. Taste the broth and add more salt or pepper if needed, and stir in a splash of hot water if the soup looks too thick.
Spoon about ½ cup of rice into each of 6 bowls and ladle the hot soup over the top. Finish each bowl with about 2 tablespoons of shredded cheddar and a sprinkle of fresh parsley, then serve right away.