How to Make It
Process yogurt, parsley, lemon zest and juice, and garlic in a blender until smooth, about 2 minutes. Transfer half of mixture to a small bowl; stir in 1 tablespoon oil and ½ teaspoon salt and set aside. Transfer remaining mixture to a dish; add chicken, turn to coat, and let stand at room temperature for 15 minutes. Remove chicken from marinade (discard marinade) and season with remaining ½ teaspoon salt.
Preheat grill to high (450°F to 500°F) and lightly oil grates. Grill chicken, covered, flipping once, until grill marks appear and a thermometer inserted in thickest portion registers 165°F, about 6 minutes per side. Transfer to a cutting board and let rest for about 10 minutes.
Meanwhile, toss peppers, onion, and remaining 1 tablespoon oil in a large bowl. Grill peppers and onion, uncovered, until charred and softened, 5 to 7 minutes. Remove from grill. Grill pitas, covered, flipping once, until warmed through and slightly toasted, about 2 minutes per side.
Slice chicken and coarsely chop onion. Serve chicken on greens with peppers, onion, reserved yogurt mixture, and pitas.