Heat the olive oil in a 26 cm / 10-inch oven-safe nonstick skillet over medium-low heat. Add the sliced onions and 1/2 tsp salt, and cook for 25 minutes, stirring every 3 to 4 minutes, until they're soft and glossy and pale gold at most. If the edges start to brown, lower the heat and stir in 1 tbsp water.
While the onions cook, heat the oven to 180 C / 350 F with a rack in the middle. Whisk the eggs, Parmigiano, parsley, remaining 1/2 tsp salt, and pepper in a large bowl until no streaks of white remain.
Spread the onions in an even layer and drop the heat to low. Pour the eggs over the onions and tilt the pan so the eggs flow through. Cook for 2 minutes without stirring, until the edges just begin to set.
Slide the skillet onto the middle rack and bake for 10 minutes, until the center doesn't jiggle when you nudge the pan and the top is lightly golden. An instant-read thermometer in the center reads 160 F / 71 C.
Set the skillet on a trivet and drape a towel over the hot handle. Rest the frittata for 5 minutes, then run a silicone spatula around the edge, slide it onto a board, and cut it into 4 wedges.