In a large nonstick skillet, heat oil over medium heat. Add onion and cook, stirring often, until soft and starting to brown, 5 to 7 minutes. Sprinkle sugar over onion and add mushrooms and garlic. Cook and stir occasionally, until mushrooms soften, about 8 minutes. Stir in parsley and transfer to serving plate. Season with salt and pepper to taste. Rinse and dry skillet and reheat over high heat.
Heat tortillas in hot skillet, 1 minute each side, adjusting heat so tortillas don't burn. Divide mushroom mixture among 8 tortillas, placing mixture on one side of tortilla. Sprinkle cheese evenly over mushrooms. Fold tortillas in half over filling.
Cook quesadillas in same skillet until golden and crisp on both sides and cheese is melted. Serve hot.