Cover chicken breasts with plastic wrap and pound to uniform thickness as needed.

Combine ¼ cup yogurt, 2 tablespoons oil, ½ teaspoon salt, cumin, cinnamon, and ⅛ teaspoon pepper in large bowl. Add chicken and toss to coat; refrigerate until ready to cook.

Bring 2 quarts water to boil in large saucepan. Add oat berries and 1½ teaspoons salt and cook until grains are tender with slight chew, 45 to 50 minutes. Drain oat berries, spread over large plate, and let cool completely, about 15 minutes.

Brush zucchini and bell peppers with 1 tablespoon oil and sprinkle with ¼ teaspoon salt. Clean and oil cooking grate. Remove chicken from marinade, allowing excess to drip off, and place on hotter side of grill. Place zucchini and bell peppers on cooler side of grill. Cook (covered if using gas), turning chicken and vegetables as needed, until chicken is browned and registers 160 degrees and vegetables are tender and charred in spots, 8 to 12 minutes. Transfer chicken and vegetables to cutting board as they finish cooking. Tent chicken with aluminum foil and coarsely chop vegetables.

Whisk lemon zest and juice, remaining ¼ cup yogurt, remaining 2 tablespoons oil, remaining ¼ teaspoon salt, and remaining ⅛ teaspoon pepper together in small bowl. Transfer ¼ cup yogurt mixture to large bowl, add oat berries, chopped vegetables, and parsley and toss to combine. Season with salt and pepper to taste. Serve chicken with salad, passing remaining yogurt mixture separately.