For the potato salad: Put the potatoes in a large pot and cover by 2 inches with cold water. Season the water liberally with kosher salt. Bring the potatoes to a simmer over medium/low heat and cook until fork-tender, about 20 minutes. Drain the potatoes in a colander, then cover with aluminum foil to keep warm until tossed with the vinaigrette.
For the sweet relish vinaigrette: In a large bowl, whisk together the vinegar, relish, capers and mustard. Slowly whisk in the olive oil to emulsify. Season with salt and pepper.
Add the potatoes, parsley, red onions, cherry peppers and chives to the bowl with the vinaigrette. Toss to coat, adding additional salt and pepper to taste. Serve immediately, or keep covered in the refrigerator up to overnight and toss again before serving.