Add olive oil to a large skillet over medium heat. Add potato and onion; they should be mostly covered with olive oil (add a little more oil if needed). Season with 1 ½ teaspoon kosher salt.
Cook on medium-high heat at a gentle boil for about 8-12 minutes, turning occasionally, until potatoes are just fork tender.
Drain the potatoes and onions while reserving the oil in a bowl. Transfer the drained potatoes and onions to a large bowl. Allow to cool for a few minutes.
Meanwhile, add salt and paprika to the beaten eggs and whisk to combine.
Add beaten eggs to the potato and onion and toss to coat.
Add 1-2 tablespoons of the reserve olive oil to the bottom of a 10-inch non-stick cast iron skillet over high heat. Once hot, pour the potato, onion, egg mixture into the pan and cook on high heat for 1 minute. Reduce heat to medium-low and cook for about 5 minutes.
During this time, gently slide a heatproof silicone spatula along the outer edges and the bottom of the tortilla to prevent it from sticking.
Once the omelette starts to firm up around the edges and the center is slightly firm (but still soft and maybe a little runny), place a plate larger than the size of the pan over the pan, hold the handle of the pan with an oven mitt, and very carefully flip the omelette onto the plate.
Now the cooked side of the tortilla de patatas is facing up. Using the spatula, gently slide the omelette back into the pan.
Cook on high heat for 1 minute, then reduce heat to low for 3-5 more minutes or until the center is done. To check, insert a toothpick into the center of the omelette. If it comes out clean, it's done. If it has wet omelette attached to it, it needs more cooking time.
Once done, very carefully flip the cooked tortilla de patata back onto a large plate. Garnish with parsley or green onions. You can serve this warm or at room temperature.
Notes
Makes enough for approximately 10-12 slices. Storage: store leftovers in an airtight container for up to 4 days.