Wash and dry the red potatoes thoroughly, then cut them into 1-inch chunks for even cooking on the grill. In a separate bowl, combine the dried rosemary, garlic powder, smoked paprika, salt, black pepper, and dried oregano. I like to crush the dried herbs slightly between my fingers before mixing them—this releases more of their essential oils and gives you better flavor distribution throughout the potatoes.
Place the cut potatoes in a large bowl and drizzle with olive oil, then add the spice mixture from Step 1 and toss everything together until the potatoes are evenly coated. Lay out a large piece of heavy-duty foil (or double layer regular foil) and mound the seasoned potatoes in the center. Fold the foil up around the potatoes to create a packet with some air space inside—this allows steam to circulate and cook the potatoes evenly. Spray the inside of the packet lightly with cooking oil to prevent sticking.
Preheat your grill to 375-400°F (medium-high heat). Place the foil packet directly on the grill grates and cook for 15 minutes without opening. Carefully open the foil packet (watch for steam), give the potatoes a gentle stir and flip them to ensure even cooking, then reseal the packet loosely and grill for another 5-10 minutes until the potatoes are fork-tender. I find that leaving the foil slightly open during this final phase helps any excess moisture escape, giving you potatoes that are tender but not waterlogged.
Transfer the grilled potatoes to a serving dish. While still warm, sprinkle with grated parmesan cheese and fresh chopped parsley, then drizzle lightly with fresh lemon juice. Toss gently to distribute the toppings evenly and serve immediately while the potatoes are still hot.