In a large pot over medium heat, heat oil. Add beef, onion, and carrot; season with salt and pepper. Cook, breaking up beef with a wooden spoon, until beef is cooked through and vegetables are just tender, about 8 minutes. Add garlic and cook, stirring, until fragrant, about 1 minute more.
Add tomatoes, cabbage, broth, rice, Worcestershire, vinegar, brown sugar, paprika, oregano, and 1 c. water; season with salt and pepper. Bring to a boil, then reduce heat to medium-low and simmer, stirring occasionally, until rice and veggies are tender, about 20 minutes. Stir in parsley.
Divide soup among bowls. Top with more parsley.
