Preheat the oven to 450°F (230°C). Meanwhile, in a small bowl, mix ¼ cup (2 fl oz/60 ml) of the oil, 2 tablespoons of the parsley, the green onion tops, and ¼ teaspoon salt to make a sauce.
In a food processor fitted with the coarse grating blade, grate the potatoes. Transfer to a bowl and mix in the remaining 3 tablespoons parsley and the white green onion parts.
Working quickly, warm 2 tablespoons oil in a 12-inch (30-cm) nonstick frying pan over medium-high heat. Add about ½ cup (3 oz/90 g) of the potato mixture to the pan and flatten with the back of a spatula into a thin pancake. Repeat, forming 3 more pancakes. Cook until brown and crisp on the first side, about 6 minutes. Sprinkle with salt and pepper, turn over, and cook until brown and crisp on the second side, about 6 minutes. Transfer the potato pancakes to paper towels to drain. Add more oil to the pan and repeat, forming more pancakes. Transfer to paper towels to drain. Transfer the potato pancakes to a baking sheet. Reheat the pancakes in the oven while you poach the eggs.
Place 2 potato pancakes on each of 2 warmed plates. Top with the eggs and spoon the green onion sauce over the top. Serve right away with the yogurt and any remaining pancakes on the side.