Heat the olive oil in a large soup pot over medium-high heat. Add the onion and sauté until translucent.
Add the vegan beef to the pot, season with salt, and cook until browned, breaking it apart with a wooden spoon as you go.
Add the carrots, garlic, and cabbage to the pot and sauté until the cabbage has wilted slightly and the carrots have softened a bit.
Stir in the broth, tomato sauce, diced tomatoes, brown sugar, oregano, pepper, bay leaves, and rice. Bring the soup to a boil and then reduce the heat to low, cover, and simmer for 30 minutes.
Taste the soup and season with salt and pepper if desired.
Serve warm in bowls garnished with fresh parsley.
Notes
To store: Transfer leftover cabbage roll soup to an airtight container and refrigerate for up to 5 days. Warm it up in the microwave or on the stovetop over medium heat. To freeze: If you want to freeze this soup, it will keep for up to 3 months in the freezer. When you're ready to enjoy, thaw it overnight in the refrigerator, then reheat according to the instructions above.