Cook a hard-boiled egg according to Basic Recipes: How to Boil the Perfect Egg. Peel and coarsely chop. Set aside for later. (You can cook the egg while you do the rest of the steps, but be sure to keep track of the time.)
For potatoes:
Scrub potatoes, removing eyes, bad spots, and any green areas. Place in a large saucepan and add enough water to cover the potatoes. Add vinegar and salt.
Cover with lid and bring to a boil over high heat. Turn down to medium-low and simmer uncovered about 15 minutes, or until a fork can just begin to pierce through the potatoes. They should still be quite firm but not crisp.
Remove from heat and rinse in a colander with cold water. Set aside to cool somewhat.
For vegetables:
While potatoes cool, chop celery, pickles, and optional parsley, placing into the large saucepan. Thinly slice scallions to yield about 1/4 cup, using both white and green parts and add to other vegetables.
When the potatoes have cooled enough to handle, cut into 1-inch chunks and chop egg. Combine everything in the saucepan. (Alternatively, set chopped egg aside to use as a garnish.)
To assemble:
Mix dressing ingredients in a small bowl. Pour over the potato mixture and toss to coat well.
Optional: Instead of mixing egg into the salad, sprinkle it on top as a garnish.