Put the potatoes and eggs into a big saucepan of cold salted water. Bring to a simmer. After 12 minutes remove the eggs with a slotted spoon and let cool. Continue cooking the potatoes until a paring knife poked into them goes in without resistance, about 3 minutes longer. Drain the potatoes in a colander and let them cool.
Reserve some scallion greens and capers for garnish. Meanwhile, stir together the mayonnaise, mustard, pickles and their juice, onion, remaining scallions and capers, parsley, and lemon juice in a bowl large enough to hold the potatoes. Peel the cool eggs and grate them into the bowl. Stick a fork into the potatoes and lift them 1 at a time out of the colander. Break up the potatoes by hand into rough chunks, add them to the bowl and toss to coat with the dressing. Season, to taste, with salt and pepper. Drizzle with a little olive oil before serving.