Wash potatoes and cut them into thick long wedges. Place in a large pot of cold water and let soak for 3 minutes. Drain, rinse well, and fill the pot back up with cold water, making sure to cover the potatoes by at least 2 inches. Add 1 teaspoon of salt to the water and stir well.
Place the pot with the potatoes over a high heat burner. Bring the water to a boil. Let the potatoes boil for 8 minutes.
Drain the water from the potatoes and place them on a cutting board to let them dry completely.
In a large mixing bowl, combine ground black pepper, garlic powder, onion powder, Italian seasoning, smoked paprika, salt, lemon juice, ground mustard, dried parsley, and olive oil. Use a whisk to mix.
Add the potatoes to the bowl and gently toss in the olive oil marinade.
Heat your grill to medium heat, about 375°F, and brush the grill grate with olive oil.
Carefully place the potatoes on the grill. Close the grill cover and let cook for 5 minutes.
Lift the grill cover and use a spatula to flip each potato over to the other side. Close the grill cover and let cook for 5 additional minutes.
Remove the potatoes from the grill and place them back into the leftover olive oil marinade bowl. Pour the melted butter over the potatoes. Sprinkle 2-3 tablespoons of finely chopped parsley over the top of the butter and toss the potatoes gently to combine.
Serve immediately.