In medium bowl, beat eggs with ¾ teaspoon salt; set aside. In large bowl, toss potatoes with ¼ teaspoon salt.
In 10-inch nonstick skillet, heat oil on medium. Add potatoes; cook 10 to 12 minutes or until tender but not falling apart, gently turning occasionally. With slotted spoon, transfer potatoes back to large bowl.
To skillet, add onion; cook 12 minutes or until very tender, stirring occasionally. With slotted spoon, transfer onion to bowl with potatoes. When potato-onion mixture has cooled slightly, gently stir in eggs until well combined.
Drain all but 2 teaspoons oil from skillet (reserve oil for another use); heat on medium-high 1 minute. Add egg-potato mixture to skillet, spreading in even layer; reduce heat to medium. Cook 7 minutes or until eggs are mostly set and edges are browned.
Loosen edges with rubber spatula. Remove skillet from heat and cover with large plate; carefully invert, holding plate and skillet together. Slide omelet back into skillet. Cook on medium 3 minutes or until bottom and center are set. Serve warm or at room temperature, garnished with parsley.
