Heat a grill to medium-high (about 450°). Rub steak with 1 tbsp. oil, then sprinkle with 2 tsp. salt and 1 tsp. pepper. Grill, turning occasionally, until charred and internal temperature registers 125° for mediumrare, 10 to 12 minutes. Transfer steak to a cutting board. Let rest 10 minutes, then thinly slice against the grain.
Meanwhile, combine 1 cup parsley, the garlic, and 3 tbsp. oil in a food processor. Process until parsley and garlic are finely chopped, about 10 seconds. Scrape parsley mixture into a bowl and stir in yogurt, lemon juice, 1/2 tsp. salt, and remaining 1/4 tsp. pepper.
Toss remaining 1/2 cup parsley, 1/2 tbsp. oil, and 1/4 tsp. salt in a medium bowl. Spread yogurt sauce on 4 plates (about 3 tbsp. each). Top with sliced steak and seasoned parsley, and serve with flatbread.