Heat 2 tablespoons of the oil in a large frying pan, add the onions and cook until soft. Add the sugar and cook for 3 minutes or until caramelized. Remove the onions with a slotted spoon and set aside. Heat the remaining oil in the pan, add the mushrooms and cook for 3 minutes or until golden brown. Set aside.
Heat a nonstick frying pan and add 1 tortilla. Scatter over a quarter of the red onions, mushrooms, Cheddar and parsley. Season to taste with salt and pepper. Cover with another tortilla and cook until browned on the underside. Turn over and cook until browned on the other side. Remove from the pan and keep warm.
Repeat with the remaining tortillas and ingredients. Cut into wedges and serve with a salad.