Fill a large bowl with cold water. Peel the Yukon Gold potatoes, cut them into 1-inch (2.5 cm) cubes, and drop them into the cold water as you work to prevent browning.
Place the eggs in a single layer in a medium saucepan, cover with cold water by 1 inch (2.5 cm), and bring to a full boil. Turn off the heat, cover, and let sit 12 minutes. Meanwhile, prepare an ice bath in a medium bowl.
Drain the potatoes, place them in a large stockpot, and cover with cold water by 1 inch (2.5 cm). Stir in 1 tablespoon kosher salt, bring to a boil, then lower to a gentle simmer and cook 10 to 12 minutes until a fork slides in easily but the cubes hold their shape.
Move the eggs to the ice bath and cool for at least 10 minutes. Peel under a thin stream of cold water, chop into 1/4-inch (6 mm) pieces, and refrigerate until assembly.
Drain the potatoes and let them steam for about 1 minute. Spread in a single layer on a rimmed baking sheet, drizzle with 2 tablespoons apple cider vinegar, toss gently, and cool about 30 minutes until no longer hot to the touch.
Finely dice the celery to get about 3/4 cup (90 g). Finely dice the red onion to get about 1/2 cup (75 g), rinse under cold water for 20 to 30 seconds, and pat dry. Chop the parsley.
In a large glass mixing bowl, whisk together the mayonnaise, yellow mustard, sweet pickle relish, celery seed, 1 teaspoon kosher salt, and black pepper until smooth. Stir in the celery, red onion, and parsley.
Add the cooled potatoes to the dressing and fold gently with a silicone spatula until every cube is coated. Fold in the chopped eggs last, then taste and adjust salt and pepper.
Cover and refrigerate for at least 2 hours. Stir gently before serving, fold in 1 to 2 tablespoons extra mayonnaise if dry, sprinkle with paprika, and serve cold.