Crack the eggs into a large bowl and mix in salt. Then mix in the tuna and cheese.
Heat 2 tablespoons of the oil in a small sauté pan over a medium flame. When the pan is hot, add the egg mixture and stir briskly a few times with a wooden spoon to prevent the eggs from sticking. Shake the pan in a circular motion for 10 seconds to keep the mixture loose as the eggs start to cook. Lower the temperature and cook for one minute more.
Place a plate over the pan and invert the pan and plate together so the tortilla winds up on the plate, raw side down. If the pan looks dry, add the remaining 2 tablespoons of olive oil.
After inverting the half-cooked tortilla onto the plate, place the peppers flat in the pan, and sear for 30 seconds on both sides. Place the tortilla back on top of the peppers so that the uncooked egg forms around the peppers.
Return the tortilla to the pan by sliding it back in, raw side down. Cook for another 60 seconds, until the bottom has fully set. To us, the perfect tortilla is very creamy and slightly runny in the middle.
Slide the finished tortilla onto a serving plate, garnish with a few sprigs of fresh parsley, and serve warm, at room temperature, or cold.