Boil a large pot of salted water. Core the cabbage and submerge it. Peel off softened leaves with tongs as they loosen; drain well.
Trim the thick rib on each leaf by shaving it flat so the leaf stays intact and flexible.
Warm olive oil in a skillet over medium heat. Cook onion until translucent, about 5 minutes. Stir in garlic for 30 seconds. Let cool.
Mix the filling: in a bowl combine ground beef, uncooked rice, cooled onion-garlic mixture, half the parsley, 1 tsp salt, pepper, paprika, and oregano.
Whisk the sauce: in another bowl combine tomato sauce, diced tomatoes, tomato paste, beef broth, brown sugar, vinegar, Worcestershire, and a pinch of salt and pepper.
Spread about 1 cup of the sauce in the bottom of a heavy pot or Dutch oven.
Fill and roll: place a cabbage leaf on a board, add about 1/3 cup filling near the stem end, fold sides in, and roll up firmly.
Arrange rolls seam-side down in the pot. Pour remaining sauce evenly over the top.
Bring to a gentle simmer over medium heat. Cover, reduce to low, and cook until rice and beef are cooked through, 75–80 minutes.
Rest 10 minutes. Spoon extra sauce over the rolls and finish with the remaining parsley.