Place the potatoes in a large pot, cover with cold water by 2 inches and add 2 tablespoons salt. Bring to a boil and cook until a knife inserted into the center of the potato meets no resistance, about 15 minutes. Drain and let cool slightly.

Whisk together the vinegar, mayonnaise and mustard in a medium bowl and season with salt and pepper. Slowly whisk in the olive oil until emulsified. Fold in the crab, half the green onions and the parsley. Season with salt and pepper.

Heat a grill to medium-high.

Slice the potatoes in half lengthwise, brush the flesh with canola oil and season with salt, pepper and little cayenne. Put the potatoes on the grill, flesh-side down, and grill until golden brown, about 5 minutes. Remove the potatoes to a platter. Break up the flesh a bit with a fork. Immediately spoon the dressing over the potatoes and garnish with the remaining green onions.