Place the potatoes in a stockpot and cover with cold water. Bring to a boil, season the water with salt, and cook until almost tender, about 15 minutes. Drain the potatoes, and when cool enough to handle, cut each potato in half.
Preheat the grill to medium. Add the mayonnaise, garlic, lemon juice, and zest to a blender and blend until smooth. Season with salt and pepper, to taste. Set aside.
Brush the potatoes with oil on all sides and sprinkle with salt and pepper. Grill, cut side down, until golden brown and just cooked through. Remove the potatoes to a platter. Drizzle the aioli over the potatoes and sprinkle with chopped parsley.