Bring a pot of water to a boil in a Dutch oven, then add red potatoes and cook until tender, about 15–20 minutes, until a knife slides in easily. Drain the potatoes and spread them on a sheet pan to cool completely for better texture.

In a bowl, whisk mayonnaise, Dijon mustard, white wine vinegar, salt, and pepper until smooth. Continue whisking until the dressing looks glossy and evenly combined.

In a large bowl, combine the cooled red potatoes, celery, green onions, and fresh parsley. Toss gently so the vegetables distribute throughout the potatoes.

Pour the creamy dressing over the potato mixture and toss well to coat every cube. Keep tossing until no dry potato pieces remain.

Refrigerate the potato salad for 2 hours before serving, until cold and set. Serve chilled and garnish with extra parsley if desired.