Brown the beef. Heat the olive oil in a large, heavy-bottomed pot over medium-high heat. Add the ground beef and let it cook undisturbed for about 2 minutes before breaking it into small pieces. Continue cooking until browned, about 7–9 minutes total. Spoon off excess grease if needed, leaving about 1 tablespoon in the pot.
Soften the vegetables. Reduce the heat to medium and stir in the diced onion and carrots. Cook for 5–6 minutes, stirring occasionally, until the onion softens. The carrots will finish cooking as the soup simmers.
Build the tomato base. Stir in the tomato paste and cook for 1–2 minutes. Add the garlic, sweet paprika, and thyme, then stir for about 30 seconds, until fragrant.
Simmer the cabbage. Add the chopped cabbage and beef broth, gently loosening any browned bits from the bottom of the pot. Stir in the bay leaves, salt, and black pepper. Bring to a gentle boil, then reduce the heat, cover, and simmer for 10 minutes.
Add the rice and tomatoes. Stir in the rinsed rice, crushed tomatoes, and Worcestershire sauce. Return to a gentle simmer, cover, and cook for another 20 minutes, or until the rice and cabbage are tender. Stir occasionally, reaching the bottom of the pot to keep the rice from sticking.
Finish and serve. Remove the bay leaves and stir in the sugar. Taste the broth and adjust the salt if needed. Stir in the fresh parsley, remove from the heat, and let the soup rest for 5 minutes. Serve with sour cream, extra parsley, and bread if you like.