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Place potatoes in a large pot. Add water to cover by 2 in, season with salt, and bring to a boil. Reduce heat to medium and cook until potatoes are tender when pierced with a knife, 20–30 minutes. Drain. Place potatoes in a large bowl and let cool slightly.
Meanwhile, whisk mayonnaise, pickle juice, Dijon mustard, sugar, 1/4 tsp. pepper, and 1 1/4 tsp. salt in a small bowl for dressing.
Using a large wooden spoon, coarsely smash potatoes, leaving some larger pieces mixed with some well-mashed pieces.
Add dressing and egg yolks to potatoes and toss to coat, coarsely smashing egg yolks. Add onion and parsley; gently mix to incorporate. Season to taste with more salt and pepper, if desired. DO AHEAD- Can be made 1 day ahead. Cover and chill.
Divide potato salad among small paper cups or bowls; dust with paprika. Top each serving with a pickle chip.
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