Combine the shallots in a large bowl with the vinegar, sugar, and salt. Set aside so they pickle slightly while you cook the potatoes.
Put the potatoes in a large pot and cover with water by 1 inch. Season with salt. Bring to a simmer and cook until tender, about 10 minutes. Drain well and let sit in the colander about 10 minutes to cool slightly.
Whisk the mustard into the shallot mixture. Season with pepper. Whisk in the olive oil in a slow, steady stream to make a slightly thick dressing. Add the warm potatoes and toss well. Add the parsley and toss again. Season with more salt and pepper, if needed.