Remove the calamari steaks from the refrigerator half an hour before cooking to allow then to come to room temperature. (If you have purchased them frozen, defrost per package instructions).
Heat a cast iron or heavy grill pan over high heat until extremely hot. If cooking outdoors, let the grill become very hot. Calamari steaks need to be cooked quickly or they can toughen.
Meanwhile, thoroughly pat the calamari steaks dry with paper towels, which will help to provide a nice sear. In a medium bowl, toss the calamari with a drizzle of olive oil until lightly coated. Don’t season the calamari until after they are cooked, as salt would draw out moisture causing then to steam rather than sear.
Grill the calamari 1 to 2 minutes per side, until lightly browned on both sides. Transfer to a platter and season with salt and pepper. Drizzle fresh olive oil on top, garnish with parsley, and serve with lemon wedges.