Place the potatoes in a large pot of cold water, add 2 tablespoons of salt and cook until potatoes are tender, but still firm, about 12 to 15 minutes. Drain, let cool and cut each potato into 3/4-inch thick slices.
Heat the grill to high.
Brush the tomatoes with the some of the oil and season with salt and pepper. While the potatoes are cooking, place the tomatoes on the grill and grill until charred on all sides and soft, about 10 minutes. Remove from the grill when charred.
Place the tomatoes, mayonnaise, garlic, paprika and chipotle in a food processor and process until smooth, season with salt and pepper. Can be made 1 day in advance and refrigerated.
Brush the potato slices on both sides with some of the oil and season with salt and pepper. Place the potatoes on the grill, close the cover and grill until lightly golden brown and just cooked through, about 2 minutes per side. Carefully transfer the potatoes to a platter and garnish with the parsley. Serve the aioli drizzled over the potatoes or on the side.
