Place the potatoes in a large pot and cover with cold water. Season with salt. Bring to a boil over high heat, then lower to a simmer and cook until tender, about 15 minutes. Drain and let rest until cool enough to handle.

In a large mixing bowl, whisk together the mayonnaise, vinegar, mustard, and salt and pepper. Add the green onions, parsley, and celery. Slice the potatoes in half, and add them to the bowl. Gently toss to combine. Taste, and add more salt and pepper if needed. Serve warm, or at room temperature.