Lay 4 sheets of aluminum foil on a work surface; place 1 crab leg on each sheet. Squeeze 1 lemon quarter on each crab leg; place squeezed lemon quarter next to each crab leg. Cut 1/4 cup butter into small cubes; dot crab legs. Sprinkle parsley and chopped garlic on each crab leg. Season each crab leg with sea salt and freshly ground black pepper.
Cover each crab leg with remaining sheets of aluminum foil; roll top and bottom edges together to create sealed packets.
Cook packets on preheated grill until crab legs are bright pink on the outside and the meat is opaque, about 20 minutes.
Melt remaining butter in a small saucepan over low heat; serve with crab.