Preheat your oven to 375°F (190°C) and lightly grease a 9x13-inch baking dish. Bring a large pot of salted water to a boil for the pasta.
Cook the penne pasta for 2 minutes less than the package directions indicate – you want it slightly underdone (al dente) since it will continue cooking in the oven. Drain well but don't rinse.
Heat olive oil in a large skillet over medium heat. Add the diced onion and green pepper, sautéing until they begin to soften, about 3-4 minutes. Add the minced garlic and cook for another 30 seconds until fragrant.
Add the ground beef to the vegetables, breaking it apart with a wooden spoon as it cooks. Brown the meat thoroughly until no pink remains, about 5-7 minutes. If excess fat accumulates, drain it off, leaving just a tablespoon for flavor.
Stir in the tomato sauce, tomato paste, brown sugar, Worcestershire sauce, mustard, paprika, salt, and pepper. Simmer the mixture for 8-10 minutes, allowing the flavors to meld and the sauce to thicken slightly.
Combine the cooked pasta with about ¾ of the sloppy joe mixture in your pasta pot, stirring gently to ensure every piece gets coated. Transfer half this mixture to your prepared baking dish, sprinkle with 1 cup of cheddar cheese, then add the remaining pasta mixture.
Top the casserole with the remaining sloppy joe mixture, spreading it evenly. Sprinkle with the remaining cheddar cheese and Parmesan. In a small bowl, mix the panko breadcrumbs with melted butter and scatter over the cheese.
Bake uncovered for 20-25 minutes until the cheese is bubbly and the top is golden brown. For an extra-crispy top, broil for the final 2 minutes, watching carefully to prevent burning.
Let the casserole rest for 5-10 minutes before serving. Garnish with fresh parsley if desired.