Heat the vegetable oil in a Dutch oven or large soup pot over medium-high heat.
Add the ground beef, season it with salt and black pepper, and let it sit briefly against the hot pot before breaking it into crumbles.
Cook the beef until no pink remains and a few browned bits form on the bottom of the pot.
Add the diced onion and minced garlic, then cook for 2 to 3 minutes, stirring often, until the onion begins to soften.
Stir in the chopped cabbage, carrots, tomato sauce, beef broth, uncooked white rice, bay leaf, and brown sugar.
Scrape the bottom of the pot while stirring so the browned beef residue dissolves into the broth.
Bring the soup to a gentle simmer, then reduce the heat to medium-low and cook for about 25 minutes, stirring once or twice, until the rice and cabbage are tender.
If the rice needs more time, simmer for another 3 to 5 minutes and add a splash of broth if the soup becomes too thick.
Remove the bay leaf, stir in the chopped parsley, and adjust the salt and black pepper before serving.