Preheat oven to 375°. In a large skillet over medium-high heat, heat oil. Add beef; season with 1 teaspoon salt and ½ teaspoon pepper. Cook, breaking up beef with a wooden spoon, until crumbled and no longer pink, 6 to 8 minutes. Using a slotted spoon, transfer beef to a plate. Pour off all but 1 tablespoon fat.
In same skillet over medium heat, cook onions, bell peppers, and garlic, stirring occasionally, until vegetables are softened, 10 to 12 minutes; season with salt and pepper. Add ketchup and cook, stirring occasionally, until ketchup is reduced to a thick paste, 5 to 6 minutes. Add chili powder and mustard and cook, stirring, until fragrant, about 30 seconds. Add tomato sauce, broth, vinegar, Worcestershire, and brown sugar and cook, stirring, until combined. Add beef and a few dashes of hot sauce (if using).
Transfer to a 2-quart baking dish. Top with rolls cut sides down (both the tops and the bottoms, for a total of 16 halves). Brush with butter and sprinkle with cheese.
Bake casserole until rolls are toasted and cheese is melted, 10 to 15 minutes. Sprinkle with parsley (if using).
