Scrub mussels in shells under cold running water. Combine 4 quarts cold water and 1/4 cup salt; add mussels. Soak for 15 minutes; drain and rinse. Discard water. Repeat soaking, draining, and rinsing twice.
For sauce, in a small saucepan melt butter over medium heat. Add garlic; cook for 1 minute. Add wine; bring just to a simmer. Remove from heat; stir in parsley.
Tear off a 36x18-inch piece of heavy foil; fold foil in half to make an 18-inch square. Place the mussels in the center of the foil. Bring up opposite edges of foil and seal with a double fold. Fold remaining edges together to completely enclose mussels, leaving space for steam to build.
Place foil packet on the grill rack directly over medium-high heat; grill for 8 to 10 minutes or until shells open. Transfer mussels (discard any that do not open) and any liquid to a serving bowl.
Transfer the sauce to small bowls for dipping. Serve mussels with sauce and, if desired, bread. Makes 4 appetizer servings.
1 1/2 pounds fresh mussels (about 20)
1/3 cup butter
1 clove garlic, minced
2 tablespoons dry white wine
2 tablespoons snipped fresh parsley