Preheat oven to 450°F (230°C). Heat oil in a large cast-iron skillet over medium-high heat. Season chicken thighs generously with salt and pepper; lay into skillet, skin-side down, and cook until chicken skin is golden brown and crispy, 8 to 10 minutes. Flip chicken and add mushrooms and vinegar to skillet, stirring the mushrooms between the chicken pieces to coat in chicken fat and vinegar; season with salt and pepper. Transfer skillet to preheated oven and bake until an thermometer inserted into the thickest part of a thigh reads 165°F(75°C), 10 to 15 minutes. Divide chicken between 4 plates. Stir butter into mushrooms to melt; spoon mushrooms and sauce over chicken and garnish with parsley to serve.