Heat oil in a large heavy pot over medium-high heat. Add ground beef; cook, stirring often, until the meat is cooked through and starting to brown slightly, 6 to 7 minutes. Add cabbage, onion, carrots and celery; cook, stirring often, until the onion is translucent, about 5 minutes.
Add brown sugar, paprika, salt, pepper and cayenne to the beef mixture; cook over medium-high heat, stirring constantly, until the spices are toasted, about 1 minute. Stir in tomato sauce and broth, scraping the bottom of the pot with a wooden spoon to release any browned bits. Bring the soup to a boil over medium-high heat. Stir in rice. Reduce heat to low; cover and cook until the rice is tender, 30 to 35 minutes. If desired, sprinkle with parsley before serving.
