Heat a large Dutch oven over medium heat. Add 3 tablespoons of oil, the chicken pieces, 1 chopped onion, paprika, 1 teaspoon of salt, and 1/2 teaspoon pepper. Brown the chicken on all sides, stirring the onions as you turn it, approximately 10 minutes.
Then add enough water to the pot to almost cover the chicken. Cover the pot, turn the heat down, and allow the chicken to slowly simmer for about 45 minutes. The chicken should be falling off the bone.
Meanwhile, add 1 tablespoon oil to a medium-sized pot over high heat. Add the second chopped onion to the pot and sauté for 2 minutes. Then add 2 cups of rice, and stir into the onions. Allow the rice to sauté another 1-2 minutes, then add the chicken stock. Bring to a boil and stir. Reduce the heat to a simmer, then cover. Allow the rice to cook according to package directions, usually 20 minutes. Once it has absorbed all the stock, stir in the parsley. Remove from heat, cover, and set aside.
Once the chicken is tender, remove it from the Dutch oven and cover it with foil to keep it warm. Dump the sour cream into a medium-sized glass bowl. Whisk ¼ cup of flour into the sour cream until well combined.
You should have approximately 3 cups of juices left in the Dutch oven—if it looks like much more than that, dump some out until it measures 3 cups. Now take 1 ½ cups of the reserved pan juices and slowly whisk it into the sour cream mixture. This ensures that the sour cream will not curdle when you heat it, so your sauce will be smooth.
Now pour the sour cream mixture back into the pot and whisk well. Keep the sauce over medium heat until it bubbles and thickens to a gravy consistency. Salt and pepper to taste.
Add the chicken back to the pot to coat. Serve the chicken and extra paprikas sauce over the rice.