Place potatoes in a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low; simmer until tender, about 25 minutes. Drain. Cut potatoes into 1-inch cubes, leaving skin on. Drizzle potatoes with vinegar while warm; set aside to cool completely, about 20 minutes.

Place eggs in a saucepan, cover with water, and bring to a boil. Remove from heat; let eggs stand in hot water for 15 minutes. Remove eggs from hot water; cool under cold running water. Peel eggs; cut 2 eggs into 1/4-inch dice. Slice the remaining egg into 1/4-inch slices; set aside for garnish.

Stir together the 2 diced eggs, mayonnaise, salt, dry mustard, celery seed, and black pepper in a large bowl; stir in hot water. Stir potatoes, celery, red bell pepper, onion, green onions, pickles, and parsley into mayonnaise mixture; cover and chill in the refrigerator for 30 minutes. Sprinkle salad with paprika and arrange reserved egg slices on top before serving.