In a large pot over medium-high heat, cook the bacon until crispy, about 6-8 minutes. Use a slotted spoon to remove the bacon to a plate; reserve the drippings in the pot.
Add the ground beef to the pot and cook over medium-high heat, breaking up the meat with a wooden spoon, until the meat is no longer pink, about 5 minutes. Add the onion and cook until softened, about 3-4 minutes. Add the garlic and cook, stirring constantly, for 1 more minute. Add the tomato paste and cook, stirring constantly, for 1 more minute.
Stir in the cabbage, beef broth, tomato sauce, tomatoes, brown sugar, and bay leaf. Bring to a boil; reduce heat to low, and simmer until the cabbage is tender, about 15-20 minutes.
Stir in the rice, season with salt and pepper to taste, and simmer for about 5 more minutes. Remove from the heat, discard the bay leaf, and stir in the parsley.
Ladle the soup into bowls and garnish each bowl with reserved bacon.