Cook potatoes whole and unpeeled. If pressure cooking follow directions for your appliance. If boiling use a large pot, bring water half to ¾ way up the whole potatoes and boil about 30 minutes. Check for doneness and continue cooking until they are fork tender.
When done, drain potatoes and allow to cool enough to be able to handle them. They should be still warm in order to aborb the melted butter mixture.
While potatoes are cooking, prepare hard boiled eggs. Cover eggs in a sauce pan by 1-2 inches. Bring just to a boil. Remove from heat and allow to sit covered for 20 minutes. Drain and run cold water over the eggs to cool down. Allow eggs to chill fully before peeling.
When potatoes are cool enough to handle peel and cut into 1" dice.
Melt butter in microwave and add cider vinegar, sugar and salt. Give the final mix a few seconds in microwave to ensure the mixture is warm. Pour over warm diced potatoes and stir well.
Chill potato mixture in fridge while you finely dice the celery stalks, pimento and onion. Mince the parsley. Peel the hard boiled eggs and chop roughly.
You can mix these into the potato salad and refrigerate up to 24 hours to allow flavours to mix. Add the mayonnaise and sweet relish an hour before serving.