Place the potatoes in a large pot and cover with water by about 1 inch. Add 2 tsp. salt and bring to a boil. Reduce heat and simmer until potatoes are tender, about 15 minutes. Drain and set aside.
Meanwhile, in a large bowl, combine the vegetable oil, rice vinegar, sugar, paprika, black pepper, and remaining 2 tsp. salt. Stir in the sliced red onions and parsley and let sit at least 10 minutes.
Slice the potatoes in half or into quarters, depending on their size. (You want bite-sized pieces.) Add the sliced potatoes to the bowl with your dressing and onions and stir gently to coat.
Refrigerate for several hours or overnight for best flavor.