In a large shallow dish, whisk together 1 tbsp each of the oil an dvinegar, the Worcestershire sauce, thyme, and a pinch each of the salt an dblack pepper. Place the steak in this marinade and turn it to coat. Cover and refrigerate it for 30 minutes or up to 1 day.
Preheat an outdoor grill or a grill pan.
Place the steak on the greased grill over medium-high heat and grill for 8 to 10 minutes, turning once, for medium-rare. Continue cooking the steak to desired doneness. Remove to a plate and cover loosely with aluminum foil. Let it sit for 5 minutes.
Meanwhile, in a large bowl, toss together the lettuce, tomato, bell peppers, cucumber, and parsley. Whisk together the remaining oil, vinegar, salt, and black pepper. Drizzle over the vegetables and toss to cover.
Slice the steak into thin strips and add it to the salad. Toss to combine.