Bring a large pot of water to a rolling boil, then carefully add the chopped potatoes. Cook for about 10 minutes or until potatoes are soft enough to be pierced with a fork, but not at all mushy. Drain and rinse in cold water. Set aside to cool completely.

Separately, combine the mayonnaise, mustard, lemon juice, salt, and pepper in a small bowl and mix well to combine.

When the potatoes are fully cooled, place them in a large bowl with the celery, onion, and parsley or chives. Mix gently to combine. Add the mayonnaise mixture and toss to coat. Cover and chill for at least 2 hours. Adjust seasoning before serving.