In a 4 qt. - 5 qt. Dutch oven, add cubed potatoes and enough water to cover, about 3 quarts. Stir in 1/3 cup Kosher salt and bring to a boil. Reduce heat and cook, uncovered, just until potatoes are fork tender, about 8 - 10 minutes.
When potatoes are tender, drain. Gently rinse with cool running water. And let sit in colander to cool, at least 10 minutes.
While potatoes are cooking/cooling, stir together the minced onion, minced celery, mayonnaise, pickle relish, yellow mustard, Cajun seasoning, and black pepper.
Add potatoes to a large bowl, pour the mayonnaise mixture over the potatoes and toss gently being careful to not break up the potatoes. When potatoes are evenly coated, add eggs and fold in gently. Cover and refrigerate, at least 2 hours.
Serve with a sprinkle of paprika and minced parsley.