Scrub the potatoes and wash them well.
Full a large pot with water, add the potatoes, and when the water has come to a boil, boil the potatoes for 25 minutes.
As the potatoes are boiling, add the two cups of water and the white vinegar to a saucepan. Bring to a boil. Once it has come to a boil then it's done. Set aside.
After the potatoes have boiled and are soft, drain them and let them cool long enough so that they can be easily handled.
Peel the potatoes and set the peeled potatoes in a large pan or bowl.
Cut the potatoes in half and slice the potato halves.
Drizzle the vegetable oil over the potatoes.
Sprinkle the sugar over the potatoes.
Grate the onion over the potatoes. It can be finely grated or coarsely grated. We like it very finely grated.
In a bowl, mix the mayonnaise with a few tablespoons of the vinegar water. Keep adding a little more of the vinegar water until the mayonnaise is the consistency of a sauce.
Pour the mayonnaise mixture over the potatoes and mix well.
Garnish with parsley and carrot curls.
Cover and refrigerate.
Serve cold.