Rinse your mussels. Discard any that are already open.
Brush the cut sides of the lemon with olive oil.
Scatter the mussels directly on the grill along with the lemons, cut-side down. Close the lid of the grill and cook for 5 minutes.
Meanwhile, heat the butter and garlic in a small saucepan until the butter melts.
Remove the mussels from the grill. They should all be open — discard any that are closed. Transfer to a bowl, drizzle with the garlic butter, and toss to coat.
Transfer to shallow bowls, scatter with parsley, and serve with crusty bread and grilled lemon.