Preheat oven to 450 degrees F. Combine rice, 1 cup water, and ¼ teaspoon salt in a small pot. Bring to a boil. Cover and reduce to a simmer. Cook until the rice is tender and liquid is absorbed, about 20 minutes. Let cool slightly.
Meanwhile, in a small bowl, combine the chicken spices with 1 teaspoon salt and ½ teaspoon pepper. On a rimmed baking sheet, toss chicken thighs with 1 tablespoon oil and season with the spice mixture. Toss carrots with one more tablespoon of oil and season with salt.
Arrange the chicken and carrots in a single layer on the baking sheet. Bake about 25 minutes, until chicken registers 160 degrees F on an instant-read thermometer.
Stir the dates, pomegranate seeds, parsley, pistachio, and lemon zest and juice into the rice. Serve with chicken thighs.