Adjust oven rack to lower-middle position and heat oven to 350 degrees. Combine 1¼ cup broth, rice, and ¼ teaspoon salt in bowl. Cover and microwave until rice is softened and most of liquid is absorbed, 6 to 8 minutes.

Meanwhile, pat chicken dry with paper towels and season with salt and pepper. Heat oil in medium ovensafe saucepan over medium-high heat until just smoking. Brown chicken well,  about 5 minutes per side; transfer to plate.

Pour off all but 1 teaspoon fat from saucepan. Add onion and ¼ teaspoon salt and cook over medium-low heat until softened and lightly browned, 5 to 7 minutes. Stir in garlic and thyme and cook until fragrant, about 30 seconds. Stir in remaining½ cup broth and wine, scraping up any browned bits. Stir in parcooked rice, breaking up any large clumps, and bring to simmer. Place browned chicken skin side up on rice, cover, and bake until rice is cooked through and chicken registers 175 degrees, about 25 minutes.  

Using potholders (saucepan handle will be hot), remove saucepan from oven. Transfer chicken to serving platter and tent chicken loosely with aluminum foil. Fluff rice with fork, cover, and let sit for 10 minutes. Stir in parsley and season with salt and pepper to taste. Serve chicken with rice.