Put the potatoes in a pot of cold water, add 2 tablespoons of salt and bring to a simmer over medium heat. Cook the potatoes until tender, but still firm, about 40 minutes. The potatoes must be cooked through and fork tender, but still holding their shape. Drain and put on a cutting board. Let cool, then cut each potato lengthwise into 8 wedges.
Heat grill to medium-high.
Brush the potatoes with the canola oil and season with salt and pepper, to taste. Grill until golden brown and cooked through, about 2 to 3 minutes per side. Transfer to a serving bowl.
Heat the olive oil in a small saucepan until it begins to shimmer. Remove from the heat and immediately add the garlic, red chili flakes and parsley and let sit for 1 minute. Season the oil with salt and pepper, to taste, and drizzle over the potatoes. Sprinkle with the cheese and serve immediately.