Preheat oven to 400°F. Heat oil in a large, high-sided ovenproof skillet over medium-high heat. Add ground beef, onion and ½ teaspoon salt; cook, stirring often to break up the meat, until browned and cooked through, 6 to 7 minutes.
Add garlic and crushed red pepper; cook, stirring constantly, until fragrant, about 1 minute. Add cabbage; cook, stirring often, until mostly tender, 5 to 8 minutes.
Stir in diced tomatoes, tomato sauce, rice, vinegar, caraway seeds, paprika, 1 tablespoon parsley and the remaining ¾ teaspoon salt; spread into an even layer. Remove from heat; sprinkle evenly with mozzarella.
Bake until the cheese is melted and slightly browned, about 15 minutes. Garnish with the remaining 1 tablespoon chopped parsley. Additional reporting by Carrie Myers, M.S. and Linda Frahm
