Season and sear. Heat the oven to 375 degrees F. Pat the thighs dry with paper towels and season them all over with the salt, pepper, paprika, and garlic powder. Dry skin browns; wet skin steams. Heat the oil in a 12-inch oven-safe skillet over medium-high heat until it shimmers. Sear the thighs skin-side down for 5 minutes without moving them, then flip and sear 3 minutes more. They will not be cooked through. Move them to a plate.
Build the base. Pour off all but 1 tablespoon of fat from the skillet. Cook the onion, carrots, and mushrooms over medium heat for 6 to 7 minutes, stirring now and then, until the mushrooms give up their liquid and the onion softens. Stir in the garlic and cook 30 seconds, just until it smells fragrant.
Add the rice and broth. Stir in the wild rice, broth, and thyme. Scrape the bottom of the skillet with a wooden spoon to loosen the browned bits. Bring the broth to a gentle simmer.
Nestle and cover. Set the thighs skin-side up into the rice, tucking them in so the rice comes partway up their sides. Cover the skillet tightly with its lid or a double layer of foil. A loose cover lets the steam escape, and the rice will not finish. Bake for 50 minutes.
Check and rest. The rice should be tender with most of the grains split open, and the chicken should read 165 degrees F at the thickest part. If the rice is still firm, re-cover and bake 10 minutes more. Let the skillet rest uncovered for 5 minutes, then scatter the parsley over the top and serve straight from the pan.
