Preheat grill to medium-high, or preheat the oven to 425º.
Cut 4 large pieces of foil about 10" long. In a large bowl, toss potatoes with olive oil, garlic powder, and oregano and season with salt and pepper.
Divide potatoes between foil pieces, then fold the foil packets crosswise to completely cover the potatoes. Roll the top and bottom edges to seal them closed.
Place foil packets on the grill and cook until just cooked through, 10 to 15 minutes. (Or transfer to the oven and bake about 15 minutes.)
Unwrap the foil packets and sprinkle mozzarella and Parmesan on top of the potatoes. Fold foil back over the potatoes and cook until cheese is melty, about 3 to 5 minutes.
Top with parsley and red pepper flakes and serve warm.