Using a mandoline, cut the potatoes into paper-thin or 1/16-inch slices and place in a bowl of cold water. Soak for 20 minutes. Drain and dry well with on paper towels, with a kitchen towel or in a salad spinner.
Preheat an oven to 350°F. Line 3 to 4 baking sheets with parchment paper. Place the potatoes in a bowl and drizzle with the oil. Season well with salt. Toss well to coat the potatoes with oil, salt and pepper. In a single layer, place the potatoes on the baking sheets. Bake, turning the potatoes, until golden and crispy, 25 to 30 minutes.
Remove from the oven and sprinkle immediately with salt. On a work surface, finely chop the boquerones. Add the chopped shallots, garlic and parsley and chop all together until finely minced and there is a little texture left. Place in a bowl and add the olive oil and vinegar. Season with pepper.
To serve, if the potato chips have cooled, place all of them on a baking sheet and warm in the oven until hot, about 10 minutes. Place the potato chips on a serving platter and serve the boquerone dipping sauce alongside.