Heat coconut oil in a large saucepan over medium heat; cook and stir onion, coconut flakes, and garlic until onion is softened, about 10 minutes.
Bring a pot of water to a boil; add chard and cook just until leaves become brighter green, about 3 minutes. Transfer chard to a bowl of ice water to stop the cooking process.
Stir chard, vegetable stock, and coconut milk into onion mixture; bring to a boil. Reduce heat and simmer until soup is heated through, about 10 minutes. Top soup with Parmesan cheese; season with salt and pepper.