In a medium bowl, whisk eggs, parsley, 2 tbsp. water, and 1/2 tsp. salt until smooth and well combined. Let stand at room temperature at least 15 minutes and up to 30.
Heat oil in a large nonstick frying pan over medium heat. Add potatoes in an even layer, cover, and cook until just tender, 4 to 5 minutes. Gently stir in 1 tbsp. chopped oregano, pepper, and remaining 1/2 tsp. salt. Turn onto a plate; set aside.
Heat 1 tbsp. butter in same pan until bubbly, 2 to 3 minutes. Add half of egg mixture. Let cook undisturbed until sides are set, 1 to 2 minutes. With a rubber spatula, drag cooked sides in toward middle, letting uncooked egg run out to re-form a circle. Repeat until top is set but still slightly moist, then scatter half of potatoes over half of omelet. Flip other half of omelet over potatoes. Cook until potatoes are warm, about 2 minutes. Cut in half, then lift onto two plates and sprinkle with more chopped oregano or leaves. Repeat with remaining butter, egg mixture, and potatoes.