Cook cauliflower according to package directions. Stir in 2 tablespoons parsley, 1 teaspoon olive oil, 1 teaspoon lemon juice, and 1/8 teaspoon pepper.
Combine chickpeas, cucumber, tomatoes, onion, 2 tablespoons parsley, the red wine vinegar, 1 tablespoon olive oil, 1 tablespoon lemon juice, and ¼ teaspoon pepper in a medium bowl. Set aside.
Stir together yogurt, 2 tablespoons parsley, 1 ½ teaspoons lemon juice, the lemon zest, and salt in a small bowl.
Layer cauliflower rice, chickpea salad, and chicken evenly in each of 4 bowls or containers. Drizzle each serving with about 2 tablespoons yogurt mixture.