Do ahead: Place 3 eggs in a small saucepan and cover with cold water, covering by at least 1 inch. Bring to a boil over high heat. Once the water is fully boiling, boil eggs for 7 minutes. Remove eggs to a bowl of cold water and refrigerate at least 30 minutes (or longer) until needed. To use later, peel and chop.
Peel potatoes and cut into equal-sized chunks, just slightly larger than you want the chunks to be in your finished salad. Place in a large pot. Cover with cold water, covering potatoes by 3-4 inches. Add some salt to the water. On the stove-top, bring to a boil over high heat. Continue boiling potatoes, checking often, until potatoes are tender. *Check potatoes by piercing with a sharp knife. They are done when the knife passes through the potato with no resistance.
Drain potatoes well and place in a large bowl. Drizzle with 1 Tbsp white vinegar. Allow to cool about 30 minutes, at room temperature.
Meanwhile, chop the Additions ingredients and prepare the dressing by whisking together all the dressing ingredients until well combined.
When potatoes have cooled, add the chopped celery, onion and hard-boiled eggs to the bowl. Pour the dressing over top. Toss to combine and coat the potatoes. Garnish the top with a bit of paprika. Cover and refrigerate until thoroughly cooled before serving, about 3-4 hours, at least. Garnish with fresh parsley to serve, if desired.
Potato salad will keep well, covered, in the fridge for up to 5 days.
Be sure to read the “Ingredient and Cook's Notes" included with this recipe’s post, where I share more detailed tips, variations and substitution suggestions for this recipe!