Preheat oven to 425°F or grill to medium-high. Coat 4 to 6 sheets of heavy duty aluminum foil with nonstick cooking spray.

Combine sausage, potatoes, green beans, corn and garlic in a large mixing bowl. Drizzle butter over mixture and add paprika. Season with salt and pepper, to taste, and toss until everything is well-coated with butter. Divide mixture evenly among prepared foil sheets. Wrap up and seal foil tightly around sausage and vegetables.

Grill foil packets for 30 minutes over medium-high heat, flipping halfway through, until vegetables are tender. Alternatively, bake foil packets on a sheet pan for 20-30 minutes, flipping halfway through.

Open foil packets carefully to allow steam to escape. Top sausage and vegetables with fresh parsley and Parmesan, and serve with lemon wedges.