Boil (whole) potatoes until soft- a knife should be able to be inserted easily. Drain potatoes and let cool 20 minutes or so.
In a large bowl, combine water, vinegar, salt, pepper, sugar, parsley, onion and celery together. Set aside.
Peel warm potatoes (skin should come off really easily using a butter knife or spoon), cut into large cubes and place into vinegar mixture in bowl. Stir to coat warm potatoes in marinating liquid. Cover and refrigerate 12-24 hours.
Before serving, stir in mayonnaise and any extra ingredients you'd like (crumbled bacon, dill, boiled eggs, dill pickles, etc.). Keep cool until ready to serve.