Sear the Chicken: Pat the chicken thighs dry with a paper towel and season generously with salt and pepper. Heat olive oil in a large skillet over medium-high heat. Place chicken skin-side down and sear for 4-5 minutes until the skin is golden brown and crispy. Flip and sear for another 2 minutes. Transfer the chicken to the slow cooker insert. Do not discard the drippings in the skillet.
Sauté Aromatics: Reduce the skillet heat to medium. Add the chopped onion, carrots, and celery to the skillet with the chicken drippings. Sauté for 5-7 minutes, until the vegetables begin to soften. Add the minced garlic and cook for another minute until fragrant.
Deglaze and Combine: Pour the beer into the skillet to deglaze, scraping up any browned bits from the bottom of the pan with a wooden spoon. Bring to a simmer and let it reduce by about a third, for 3-4 minutes. Pour the vegetable and beer mixture over the chicken in the slow cooker. Add the chicken broth, dried thyme, and the bay leaf. Stir gently to combine.
Slow Cook: Cover the slow cooker and cook on LOW for 4-5 hours or on HIGH for 2-3 hours, until the chicken is tender and cooked through.
Cook the Rice: About 45 minutes before the cooking time is complete, stir the rinsed long-grain white rice into the slow cooker. Ensure the rice is fully submerged in the liquid. Cover and continue to cook until the rice is tender and has absorbed most of the liquid. This prevents the rice from becoming mushy.
Serve: Remove the bay leaf before serving. Fluff the rice with a fork. Serve the beer braised chicken and rice warm, garnished with fresh chopped parsley.