Prepare a iced water bath for the eggs. In a medium saucepan, combine potatoes with salted water to cover and bring to a boil. Gently add eggs, then boil 10 minutes more until potatoes are fork-tender. Transfer eggs the prepared ice bath to cool. Drain potatoes.
Whisk together mustard and vinegar in the bottom of a large bowl. Add oil in a slow and steady stream, whisking constantly, until emulsified; remove and reserve 2 tablespoons dressing for serving. Add potatoes, chives, and parsley to large bowl; toss to combine. Season with salt and pepper.
Peel eggs and quarter lengthwise. Season with salt and pepper. In a serving bowl, layer potatoes and eggs, drizzling eggs with extra dressing.