Step 1Place the potatoes in a large saucepan, barely cover with water, add 1 tablespoon salt and bring to a simmer. Maintain a gentle simmer until the potatoes can be pierced easily with a paring knife, 20 to 30 minutes, depending on their size.
Step 2While the potatoes are cooking, prepare the vinaigrette. Place the shallots in a small bowl and cover with the vinegar. Let stand until the shallots have softened slightly, about 5 minutes. Whisk in the mustard, oil and 1/2 teaspoon salt until the mixture is smooth and creamy.
Step 3Drain the potatoes. As soon as they are barely cool enough to handle, cut them into bite-size pieces. If you prefer potatoes peeled, do this first -- the skins will pull away easily with a paring knife.
Step 4Place the potatoes in a large mixing bowl. While they are still very warm, pour over the vinaigrette and toss to combine. Set aside until cool.
Step 5When almost ready to serve, add the celery, onion and parsley and stir to combine. Salt to taste and serve.