Make marinade: Stir together parsley, garlic, anchovies, salt, pepper, chile flakes, lemon zest, and oil in a small bowl. Pour 1/2 cup of it into a large resealable plastic bag. Add chicken, seal bag, and turn bag over several times to ensure chicken is completely coated. Chill at least 8 hours and up to overnight. Chill remaining marinade in a small airtight container.
At 30 to 45 minutes before serving, stir together olives, tomatoes, capers, lemon juice, and reserved marinade in a medium bowl. Let stand at room temperature until ready to serve.
Preheat a grill to medium-high, about 400°. Drain chicken well and pat dry (discard marinade). Grill chicken until deep golden and cooked through, about 4 minutes per side. Transfer to a rimmed cutting board and let rest 5 minutes.
Cut chicken into thick slices and top with accumulated juices and olive mixture.