Cook the rice according to the package directions.
Preheat oven to 450°F. Place a wire rack on a rimmed baking sheet.
Toss together the mushrooms, parsley, oil, lemon juice, 1 teaspoon of the salt, and ¼ teaspoon of the pepper in a medium bowl. Set aside, stirring occasionally, 20 minutes.
Meanwhile, stir together the mayonnaise, Parmesan, and the remaining salt and pepper in a small bowl. Spread evenly on top of the chicken breasts. Place the chicken in a single layer on the prepared rack.
Bake 12 minutes. Turn the oven to broil and continue cooking until the chicken is cooked through and golden brown, 5 minutes. Toss together the arugula and mushroom mixture and serve with the chicken and rice.