Black rice and red quinoa are available at better supermarkets and at natural foods and specialty foods stores.
Bring rice and 1 cup water to a boil in a small saucepan. Cover, reduce heat to low, and cook until water is absorbed and rice is tender, 25–30 minutes.
Meanwhile, combine quinoa, bay leaf, 1/4 tsp. salt, and 2 cups water in a medium saucepan. Bring to a boil. Cover, reduce heat to low, and simmer until quinoa is tender, about 15 minutes. Drain; return quinoa to hot saucepan. Cover and let sit for 15 minutes. Discard bay leaf, fluff quinoa with a fork, and transfer to a large bowl.
Heat 2 Tbsp. oil in a large skillet over medium heat. Add onion and cook, stirring occasionally, until soft, about 8 minutes. Add garlic and cumin and cook, stirring often, for 2 minutes. Add to quinoa. Add rice; mix well. Stir in remaining 2 Tbsp. oil, fresh lemon juice, cilantro, parsley, and chives. Season to taste with salt and pepper. Cut avocado into wedges. Serve salad with avocado and lemon wedges.