Scrub the potatoes, and cut them into roughly 1 inch pieces. Try to get them close to the same size, so that they cook evenly.
Add the potatoes to a 3 quart sauce pan, add the first amount of salt, and simmer gently over low-medium heat until a kinife inserted in a potato goes in easily. Slightly underdone is better than over cooked, so don't leave them too long. 15 to 20 minutes will probably do it.
In the meantime, chop the celery in a medium dice [1/4 inch pieces], slice the green and white part of the scallions in 1/8th inch slices, and roughly chop the olives.
Peel and chop the boiled eggs.
Mince the parsley.
When the potatoes are cooked, drain them, and rinse briefly under cold, running water.
At this point, I like to spread the potatoes out on a paper towel lined baking sheet to cool for a few minutes. *see notes This will dry the surface of the potatoes which will ensure the salad isn't soggy.
Add all of the ingredients to a bowl large enough to allow gentle mixing of everything, until it is all evenly combined.
Cover the salad, and refrigerate for at least 6 hours before serving.
Check the salad before serving, and add a bit more mayo, if it seems dry.
Notes
*If you are in a bit more of a hurry. you can set the tray of potatoes in the freezer for no more than 15 minutes to cool and dry them quickly. [this is an old restaurant trick]. Don't forget them - if they freeze, they will be ruined. This will cut down on the total cooling time however, and works well if you don't have 6 hours for the salad to chill.Leftovers absolutely must be refrigerated, for up to about 3 days.