Place eggs in a saucepan and cover with cold water. Bring water to a boil and immediately remove from heat. Cover and let eggs stand in hot water for 10 to 12 minutes. Remove from hot water, cool, and peel.
While the eggs are cooking, place the potatoes in a saucepan with just enough water to cover. Bring to a boil, and cook for 10 to 15 minutes, or until tender. Remove from heat, drain, and set aside.
Heat oil in a skillet over medium heat. Add onions, and saute until lightly browned and translucent, about 10 minutes.
Flake salmon and spread it over the bottom of a glass baking dish or serving dish. Spread the sauteed onions over the salmon. Gently spread a little bit of the mayonnaise over the onion layer. Top with potatoes, and spread enough mayonnaise just to add moisture to the layer. Finally, slice the eggs, and cover the layer of potatoes. Spread mayonnaise over the eggs, and garnish with chopped parsley. If you wish, you can reserve some of the egg to chop and sprinkle over the top as well. Chill for 1 hour before serving.