Peel potatoes a cut them into even sized pieces. In a large saucepan, cover potatoes with 1" water, bring to a boil, then add 1 Tablespoon kosher salt.
Reduce heat to medium and simmer potatoes until tender, about 8 minutes.
After draining the cooked potatoes, immediately put them in a large bowl and gently toss with vinegar. While the potatoes are cooling down, finish the dressing.
Chop the celery rib fine, chop the green onions. Mince the red onion and chop the parsley.
Place the celery, both kinds of onions, pickle relish, parsley, mayonnaise, mustard, celery seed, black pepper and 1/2 teaspoon kosher salt in a large bowl and mix until blended.
Fold dressing and chopped eggs (if using them) into still warm potatoes, gently folding until just mixed.
Cover and chill at least one hour, then check potato salad for additional seasoning if necessary. Store any leftover potato salad in an airtight container in the refrigerator.