Move an oven rack to the lowest position and place a baking sheet on the rack. Preheat the oven to 500 degrees. Using a paring knife, cut the salmon fillets into 4 equal pieces.
On the skin of each piece of salmon, make 4 shallow cuts, making sure to not cut into the pink part of the salmon.
Use a couple of pieces of paper towels to pat the salmon fillets dry. Then, rub each fillet with 1 teaspoon of olive oil and season with salt and black pepper. Take the baking sheet out of the oven and then reduce the oven temperature to 275 degrees Bake until the internal temperature of the salmon reaches 125 degrees F, about 8-12 minutes.
In a medium bowl whisk together the olive oil, lemon juice, shallot, garlic, sugar, and parsley. Season to taste with salt and pepper.
Move the salmon to a platter and spoon the sauce over the salmon fillets. Serve immediately.