Light a large chimney starter filled with charcoal and allow to burn until coals are fully ignited and partially covered with thin layer of ash, about 20 minutes.
Build a 2-level fire by arranging half of the coals over grill bottom and piling remaining coals on 1 side of grill. Position cooking grate over coals, cover grill, and heat grate until hot, about 5 minutes. Grill is ready when side with coals is hot.
Cut swordfish steaks in half to make 4 equal pieces.
Brush fish with olive oil and generously sprinkle with salt and pepper.
Scrape grate clean with grill brush. Lightly dip small wad of paper towels in vegetable oil; holding wad with tongs, wipe cooking grate.
Grill swordfish, uncovered, turning once (using a spatula, never tongs on fish), over hot part of grill until steaks are covered with dark grill marks, 6 to 7 minutes.
Move fish to cool part of grill and cook, uncovered, turning once, until center is no longer translucent, 3 to 5 minutes.
Remove from grill and place a small slice of butter on each, a spritz of lemon juice, and a sprinkle of parsley. Serve immediately.
Gas Grills:
Turn all burners to high, cover, and heat grill until very hot, about 15 minutes.
Follow recipe for steaks through oiling the grate, grilling swordfish, covered, turning once, over high heat until steaks are covered with dark grill marks, 7 to 9 minutes.
Turn heat down to medium and cook, covered, turning once, until center is no longer translucent, 4 to 6 minutes.