Make the pulao
Put the rice in a bowl and fill with cold water. Swish the rice with your fingers to release excess starch, then pour off the water. Repeat 2 or 3 more times until the water is less cloudy. Add enough cold water to cover the rice by about 1 inch, stir in 2 tsp. salt, and set aside to soak.
Meanwhile, remove and reserve the skin and most of the fat from the chicken. Pat the chicken dry. Heat the oil in a 5- to 6-quart Dutch oven (or other wide, heavy-duty pot) over medium heat until shimmering hot. Add the chicken skin and fat and cook, turning occasionally with tongs, until the skin renders its fat and becomes crisp, 5 to 8 minutes. Using a slotted spoon, transfer to paper towels to drain and cool.
Raise the heat to high and stir in 1 tsp. salt. Add half of the chicken thighs and cook, flipping once, until browned on both sides, about 6 minutes total; transfer to a plate. Repeat with the remaining chicken. Lower the heat to medium, add the onions, and cook, stirring occasionally, until softened, about 3 minutes.
Return the chicken to the pot along with the tomatoes, parsnips, and carrots. Cook, stirring often, until the tomato begins to break down and the parsnips and carrots begin to soften, about 6 minutes. Stir in 3 cups water and 1-1/2 tsp. salt, raise the heat to medium high, and bring to a boil. Lower the heat to maintain a simmer and cook, partially covered, for 5 minutes to flavor the liquid.
Drain the rice and sprinkle it over the liquid. Bring to a boil over high heat, cover tightly, then lower the heat to medium and cook for 5 minutes. Lower the heat to low and cook, covered, until the rice is just tender and the chicken is cooked through, about 25 minutes. If the rice is very wet at this point, uncover and cook over medium heat to evaporate some of the liquid, 5 minutes, and then let sit off the heat, covered, for 5 minutes. Otherwise, let the rice sit, covered, for 10 minutes. Season to taste with salt.
Make the yogurt sauce
While the rice is cooking, drain the yogurt in a cheesecloth-lined sieve set over a bowl until it’s slightly thicker and holds its shape, 10 to 20 minutes. Transfer to a medium bowl and stir in the radish, herbs, garlic, and salt and pepper to taste. Keep at cool room temperature.