Make the potato salad: Put the potatoes in a large pot and cover with cold water by 2 inches; season liberally with salt. Bring to a simmer over medium-low heat and cook until fork-tender, 15 to 20 minutes. Drain the potatoes in a colander, then cover with foil to keep warm until tossed with the vinaigrette.
Make the vinaigrette: In a large bowl, whisk the vinegar, relish, capers and mustard. Slowly whisk in the olive oil until emulsified. Season with salt and pepper.
Add the potatoes, parsley, red onion, cherry peppers and chives to the bowl with the vinaigrette. Toss to coat and season with salt and pepper to taste.