Put the potatoes in a large saucepan; cover with cold water and season with 1 teaspoon salt. Bring to a simmer and cook until tender, 10 to 12 minutes. Drain and transfer to a large bowl. Add the vinegar and 1/2 teaspoon salt; toss. Let cool slightly, about 10 minutes.
Meanwhile, put the eggs in a medium saucepan and cover with cold water. Bring to a boil, then remove from the heat and let sit, covered, 10 minutes. Drain the eggs and run under cold water until cool; peel, chop and set aside.
Whisk the mayonnaise, sour cream, pickle brine, mustard and sugar in a small bowl. Stir in the pickles, celery, scallions and parsley. Spoon the mayonnaise mixture over the potatoes and mix well. Season with salt and pepper. Add the hard-boiled eggs and gently stir to combine. Cover and refrigerate at least 2 hours or overnight. Sprinkle with paprika just before serving.
Photograph by Con Poulos