Preheat oven or grill to 375-degrees.
Place the prepared potatoes (halved and quartered) into a 13 x 9 baking dish (disposable works) that has been sprayed with non-stick cooking spray.
Add the prepared celery, celery leaves (optional), and onion to the dish.
Drizzle the olive oil over the potatoes and vegetables, and season with salt and pepper. Toss or stir to coat.
In a medium bowl whisk together the mayonnaise, heavy cream, Parmesan cheese, garlic, mustard, and dill pickle relish.
Pour the mayonnaise mixture over the potatoes in the dish and stir well to combine.
Cover the tray with aluminum foil and roast in the preheated oven, or on the outdoor grill on top of bricks for indirect cooking, for 45-50 minutes or until the potatoes are fork-tender.
Serve with additional grated Parmesan cheese, and chopped parsley if desired.