Press the parsley stems with the side of your knife to crack them open, which helps release the flavors. Then, finely chop the parsley stems.
Heat the butter in a large pan over medium heat. Add the shallot, garlic, celery, and parsley. Sauté for 2 minutes until the vegetables are soft but not browned. Season with pepper.
Deglaze with the white wine and bring to a boil. Add the mussels and cover the pan. Once the mussels start cooking, shake or stir the mussels well. Bring them back to a boil, stirring occasionally. The mussels are ready when they have opened. Remove the mussels from the pan with a slotted spoon. Make sure all the shells are open.
Bring the mussel broth to a boil and add the cream. Let the broth simmer on low heat to reduce slightly. Add the mussels back in and stir well. Cover the pan and cook for another minute. Add three-quarters of the parsley and stir through.
Serve the mussels in a deep dish and garnish with the remaining parsley leaves. Enjoy with a green salad, white bread, or fries.