Place the mussels in a large bowl of fresh water (cold / room temperature) and rinse well. Remove the beards by pulling the grassy bits from the pointy edge of the mussels towards the rounded edge. Drain the water and set aside.
In a large pot, heat the oil and butter over medium heat. Add the onion, garlic and thyme, frying until it is soft and fragrant, but not too dark. Add the wine and turn up the heat to high. As soon as the liquid comes to a boil, add the mussels all at once and cover the pot with a lid. Steam for 3 minutes, then give the mussels a stir. Cover and steam for another 2 minutes, making sure that the mussels have opened properly. Remove from the heat, add the cream and give it a final stir. Ladle into bowls, scatter with the chopped parsley and some more thyme, and serve at once with crusty bread to mop up the fragrant sauce.