Preheat the oven to 425°F.
Melt the cooking fat in a large, heavy saucepan or Dutch oven over medium-high heat. Meanwhile, season the chicken thighs on all sides with salt and pepper. Add the chicken in batches to the hot pan, skin side down, taking care to avoid crowding, and cook them for about 6 minutes or until they are golden brown; turn and repeat on the other side. Transfer each browned batch to a plate and set aside.
Add the garlic to the saucepan and cook, stirring often, until the cloves are lightly browned and beginning to soften, 2 to 3 minutes. Pour the wine into the hot saucepan and stir to scrape up the browned bits on the bottom of the pan; bring the wine to a simmer. Cook until it reduces by half, about 2 minutes. Add the broth and return the mixture to a simmer. Add the potatoes and season with salt and pepper.
Tuck the browned chicken thighs in a single layer between the potatoes. Spoon some of the cooking liquid over the chicken thighs. Scatter with the thyme springs and parsley.
Roast in the oven until the potatoes are fork-tender and the chicken is cooked through, 35 to 40 minutes.
Serve the chicken and potatoes with a drizzle of the pan juices.