Heat the oven to 350°F (175°C). Lightly coat a 9-by-13-inch (23-by-33-cm) baking dish or a large Dutch oven with cooking spray.
Bring a large pot of water to a boil. Cut around the core of the cabbage without cutting it all the way out. Place the cabbage in the boiling water and cook until the outer leaves soften enough to pull away, about 3 to 5 minutes. Remove softened leaves with tongs and repeat as needed until you have 12 large leaves. Let the leaves cool slightly. Trim the thick center rib from each leaf with a small knife so it rolls easily.
In a large bowl, combine the ground beef, uncooked rice, onion, garlic, egg, paprika, salt, pepper, and parsley, if using. Mix gently with clean hands or a fork until evenly combined; do not overmix.
Spread the drained sauerkraut across the bottom of the prepared dish. In a medium bowl, stir together the crushed tomatoes, tomato sauce, beef broth, and brown sugar, if using. Spoon about 1 cup (240 ml) of the tomato mixture over the sauerkraut.
Place about 1/4 cup (60 ml) of filling near the stem end of each cabbage leaf. Fold the sides inward, then roll up firmly, like a burrito. Place the rolls seam-side down in the dish.
Pour the remaining tomato mixture over and around the rolls. Cover the dish tightly with foil or a lid. Bake for 75 minutes. Uncover and bake for 10 more minutes, until the rice is tender and the center of a roll reaches 160°F (71°C).
Let the cabbage rolls rest for 10 minutes before serving. Spoon the tomato-sauerkraut sauce over each portion.