Heat a medium Dutch oven or heavy pot over medium-high heat. Add the beef and a hefty pinch of salt. Using a spatula, spread the beef in an even layer to cover the bottom of the pot. Sear for 3 minutes, then break the beef into smaller pieces and continue cooking for 3 minutes, stirring occasionally, until evenly browned.
Stir in the rice and cook, stirring frequently, for 3 minutes, until the rice develops a golden color
Add the onion and cook, stirring frequently, for 5 minutes, until softened and translucent.
Add the paprika and garlic and cook, stirring, for 1 minute, until fragrant
Add the tomatoes, crushing them as you add them to the pot, plus 6 cups of water, the sauerkraut and brine (or the cabbage and vinegar), the bunch of parsley and a hefty pinch of salt. Bring to a boil, loosely cover the pot with a lid and boil for 15 minutes
Taste and season with more salt if needed. Discard the parsley bunch and serve the soup with freshly chopped parsley on top
