Place potatoes and sweet potato on an 18 inch x 15 inch piece of heavy-duty foil. Drizzle with half of the butter, garlic, parsley, salt and lemon-pepper. Seal packet tightly. Grill, covered, over indirect heat for 20 minutes on each side.
Meanwhile, place mushrooms, onion, green pepper, zucchini and summer squash on a 20 inch by 18 inch piece of heavy-duty foil. Drizzle with remaining butter; sprinkle remaining seasonings. Seal packet tightly. Grill, covered over medium-hot heat for 10 minutes on each side or until veggies are crisp-tender.
Combine the contents of both packets in a bowl; sprinkle with cheese.
