Place the peeled and cubed potatoes in a large pot and cover with cold water. Add a pinch of salt to the water.
Bring the water to a boil over medium-high heat and cook the potatoes until they are fork-tender, about 12-15 minutes.
While the potatoes are cooking, place the eggs in a medium saucepan and cover with cold water by about an inch.
Bring the water to a boil over medium-high heat, then cover the pan, remove it from heat, and let the eggs sit for 12 minutes.
Drain the hot water from the eggs and transfer them to a bowl of ice water to cool for 5 minutes. Peel the eggs and chop them roughly.
Drain the cooked potatoes and transfer them to a large mixing bowl. Let them cool slightly for about 5 minutes.
In a separate bowl, whisk together the mayonnaise, Dijon mustard, apple cider vinegar, granulated sugar, salt, and black pepper until smooth.
Add the chopped celery, red onion, and parsley to the potatoes.
Pour the dressing over the potatoes and vegetables and gently fold to combine without breaking the potatoes.
Add the chopped eggs to the salad and gently fold them in to distribute evenly.
Taste and adjust seasoning with additional salt and pepper if needed.
Cover the potato and egg salad and refrigerate for at least 1 hour before serving to allow flavors to meld.