Boil potatoes and eggs in a large pot (~ 8-quart pot) of water.
While the potatoes are boiling start chopping all ingredients. Chop green onions, celery and pickles into small pieces.
After 20-30 minutes, when the eggs are boiled, remove them from the water but let the potatoes continue boiling.
Peel the boiled eggs and separate the whites from the yolks. Chop the whites of the 4 eggs into small pieces and set aside. In a small bowl, smash egg yolks with fork, add oil and whisk together vigorously to make your own mayonnaise. Add the store-bought mayonnaise and mustard and mix well. Homemade mayo and store-bought Mayo equals about 1 cup.
Dice the egg white into small pieces and set aside.
When potatoes are done pour off the boiling water. Let the potatoes cool slightly so they are easier to handle. Holding potato with a fork, and a sharp knife in the other hand, peel skin from potatoes. The skin comes off easily and does not take a lot of potato with it. Place peeled potatoes in large mixing bowl.
Cut potatoes into approximately 1-inch squares. It is desirable for many potatoes to break apart. Sprinkle potatoes with salt and stir the cooked potatoes and sprinkle more salt. Set aside to cool.
Add egg yolk/oil/mayonnaise mixture to potatoes, onions, celery, pickles and parsley. Add more mayo and mustard if needed to incorporate all the ingredients to a soft consistency but not runny.Add salt and pepper to taste.
Put the potato salad in a serving bowl, sprinkle with paprika and refrigerate.
This dish is best served cold. It is recommended to make this dish the day before it is to be served for optimal flavor.