Preheat the oven to 375 degrees F. Heat the olive oil in a 10-inch ovenproof nonstick skillet over high heat. Add the potatoes, onion, 1 1/2 teaspoons salt, and pepper to taste. Cook, turning occasionally, until the potatoes are tender, 10 to 12 minutes. Meanwhile, beat the eggs in a large bowl until foamy.
Set a colander in another large bowl and transfer the cooked potatoes, onion and cooking oil to the colander. Toss gently to drain as much oil as possible; reserve the oil in the bowl. Add the potatoes and onion to the eggs, making sure they are submerged.
Heat 1 tablespoon of the reserved cooking oil in the skillet over medium-high heat. Add the potato-egg mixture and press evenly into the pan with a rubber spatula. Cook until the underside is golden brown, about 4 minutes. Transfer to the oven and bake until set, about 8 more minutes. Loosen the edge of the tortilla with the spatula, then invert onto a plate. Season with salt. Cut into wedges.
Toss the parsley with the piquillo peppers, 1 tablespoon of the reserved cooking oil, the lemon juice, and salt and pepper to taste. Serve with the tortilla.