Cook the spuds. Quarter the potatoes (leave the skins on) and add them to a pot large enough to cover them fully with water. Add plenty of salt to the water to season the potatoes, and bring the water to a simmer. Simmer until the potatoes are soft enough to pierce through with the tip of a knife or toothpick, maybe 15 to 20 minutes. Then strain and cool them.
Mix everything else. Chop the celery, scallions, and parsley. Peel and coarsely chop the hardcooked eggs. In a large bowl, whisk the vinegar while very slowly adding the oil, whisking until the mixture thickens slightly. Add the cooled potatoes, celery, scallions, parsley, eggs, salt and pepper, and toss everything together well until all of the dressing has been absorbed and the salad is thoroughly mixed.
Season and chill. Taste the salad and adjust the seasonings to your taste preference. If you find it too tart, add a pinch of sugar to sweeten it up. Pop the salad into the fridge for an hour to allow the flavors to meld. The salad can even be made a day ahead of time to save prep time on the day of a large cookout.
Serve. Serve cold or at room temperature.