Flatten beef to 1/4-in. thickness. In a shallow dish, combine the oil, vinegar, Worcestershire sauce, Italian seasoning, garlic powder and 1 teaspoon each salt and pepper; add steak and turn to coat. Cover; refrigerate for 8 hours or overnight.
Drain and discard marinade. Combine the cheese, garlic, onion, parsley and remaining salt and pepper; sprinkle over steak to within 1 in. of edges. Roll up jelly-roll style, starting with a long side; tie with kitchen string at 1-in. intervals. Insert a skewer into each piece where it is tied. Cut into 1-1/4-in. rolls.
Lightly oil the grill rack. Grill over medium heat for 6-8 minutes on each side until meat reaches desired doneness (for medium-rare, a thermometer should read 135°; medium, 140°; medium-well, 145°-done).