In a medium saucepan, bring the rice and 2 1/4 cups water to a simmer. Cover and cook over medium-low heat until the water has evaporated and the rice is tender, about 20 minutes.
Meanwhile, season the chicken with salt and pepper. In a large nonstick skillet, heat the oil over medium-high heat. Add the chicken and cook, turning once, until browned, about 7 minutes; transfer to a plate. Add the onion, garlic and walnuts to the skillet and cook, stirring often, until the onion is softened, about 5 minutes. Stir in the cinnamon, scraping up any browned bits. Return the chicken and any juices to the pan, add the cranberry-pomegranate juice and simmer until the chicken is cooked through, 3 to 4 minutes; transfer the chicken to a serving platter.
Cook the sauce, stirring occasionally, until thickened slightly, about 3 minutes; stir in the parsley and pour over the chicken. Serve on top of the rice.