If using a white-fleshed fish, marinate in olive oil for one hour. If using an oily fish, skip to step 2 (see footnote)
Preheat a barbecue grill to medium. Combine the butter, garlic, parsley, lemon juice and salt and pepper to taste in a small bowl. Brush a little of the butter mixture over both sides of the fish cutlets.
Place a square, hinged wire grill on a piece of newspaper and brush the inside bars well with olive oil to prevent the fish from sticking to the barbecue. (Cooking fish in a hinged grill makes it easy to turn without the danger of it breaking up, and also gives the fish an attractive grid-marked finish.)
Secure the fish in the hinged grill, then lay the grill on the barbecue. Cook, basting every 2 minutes with the remaining butter mixture, for 3–4 minutes on each side, or until the fish is cooked through when tested with a fork. Serve with lemon wedges and a green salad.