Peel and cut potatoes into ¾ inch cubes.
Cover potatoes with an inch of water in a large saucepan and bring to a boil.
Add 1 tablespoon salt to boiling water and reduce heat to medium.
Simmer about 8 minutes, stirring gently once or twice, until potatoes are tender. Taste a piece—don't overcook.
Drain potatoes and place in a large bowl.
Toss potatoes gently with vinegar using a rubber spatula.
Cool potatoes. They must be no more than barely warm to mix.
Chop celery fine.
Stir together celery, celery seeds, mayonnaise, dry mustard, onion, parsley, pepper, and ½ teaspoon salt.
Gently fold together dressing mix and potatoes.
Cover and refrigerate at least an hour, until thoroughly chilled.