Place potatoes in a large pot and cover with water by 2 inches .Season the water generously with salt. Bring to a boil and cook for 10-15 minutes, until just tender. Drain.
While the potatoes are still warm, peel and slice, or cube them, into a large serving bowl.
Meanwhile, cook the sliced bacon in a large skillet over medium heat until browned, about 5-6 minutes stirring occasionally. Transfer the bacon to a plate topped with paper towels to drain. Pour out all but 2 tablespoons bacon fat from the pan.
Add the onion to the bacon fat. Season with a pinch of salt and cook for about 3-4 minutes, until softened.
Stir in 1/4 cup water, vinegar, mustard, sugar, salt and pepper.
Bring to a boil and pour over potatoes. Toss to coat. Add the bacon, celery and parsley. Season with additional salt and pepper, to taste. Serve warm or at room temperature.