Put eggs in a pot and cover with cold water. Bring to a boil, cover, take off heat, and let sit 14 minutes. Meanwhile, prepare a large bowl of ice water. Transfer eggs to ice-water bath. Let sit at least 10 minutes or up to 1 hour, until you are ready to use.
Peel potatoes; cut into bite-size pieces. Put potatoes in a large pot, cover with cold water, and bring to a boil. Add 1 tbsp. salt, reduce heat to maintain a slow boil, and cook potatoes until tender to the bite, about 8 minutes. Drain potatoes, put in a large bowl, and toss with vinegar. Let cool to room temperature, about 30 minutes.
Meanwhile, finely chop onion, rinse with cold water, and pat dry. Set aside. Finely chop celery, pickles, and parsley. In a bowl, mix mayonnaise, onion, celery, pickles, parsley, mustard, pepper, and remaining 1/2 tsp. salt. Peel and chop eggs.
Gently toss cooled potatoes with dressing. Gently mix in eggs. Serve immediately.
Note: This salad is at its best right after it's made, but it may be kept, covered and chilled, for up to 2 days.