Put the potatoes in a saucepan and cover with 2 inches of water. Generously season the water with salt. Bring to a boil, reduce the heat to medium, and simmer until tender, about 10 minutes. Drain and let stand until cool enough to handle, about 5 minutes.
Transfer the potatoes to a large bowl. Season liberally with salt and pepper and drizzle with the vinegar; toss to coat.
In a separate bowl, combine the mayonnaise, mustard, onion, chives, dill, parsley, and lemon zest. Pour the dressing into the bowl of potatoes and toss. Season to taste with salt and pepper. Cover the bowl with plastic wrap and refrigerate for at least 1 hour and up to 8 hours for best results.