Put steak in a large resealable plastic bag. Whisk olive oil, lemon juice, Worcestershire sauce, garlic, oregano, salt, and pepper in a small bowl. Reserve 1/2 cup marinade for vegetables; pour remainder over steak and turn to coat. Seal bag. Chill at least 2 hours or up to 12 hours.
Preheat oven to 450 degrees F (230 degrees C). Place 1 rack in center position and another 4 inches from broiler. Line a baking sheet with aluminum foil and spray with cooking spray.
Toss chickpeas, tomatoes, cucumber, and onion with reserved 1/2 cup marinade on the prepared baking sheet and spread in an even layer.
Roast on center rack until vegetables begin to pucker and brown, 15 to 20 minutes.
Remove baking sheet from oven and turn oven to broil. Push vegetables to the middle of the pan. Remove steak from marinade, allowing excess liquid to drip off, brush off garlic, and set on top of vegetables. Discard marinade.
Broil steak on top rack, flipping once, until it begins to char and an instant-read thermometer inserted into thickest part registers 125 degrees F for rare or 135 degrees F for medium-rare, 3 to 5 minutes per side.
Cover loosely with foil and let rest 10 minutes before slicing steak thinly across the grain. Serve warm steak and vegetables with pan juices over romaine, sprinkled with feta cheese and parsley.