Preheat a charcoal or gas grill to high heat.
Brush the potato wedges with oil, sprinkle with salt and pepper and grill until nicely charred and cooked through, 2 to 3 minutes per side.
While the potatoes are grilling, whisk together the mayonnaise, mustard, sherry vinegar and grilled lemon juice. Whisk in the olive oil. Season with salt and a generous amount of pepper.
Pull the potatoes from the grill to a platter and dust with smoked paprika. Drizzle the sauce over the warm potatoes and scatter the parsley over the top.;