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Heat a large skillet. Add the vegetable oil and heat until wisps of smoke appear. Add the chicken drumsticks and breasts in a single layer, season with salt and pepper and cook the chicken over moderately high heat, turning once, until golden brown, about 7 minutes total; transfer the chicken to a plate. Add the mushrooms to the pan and cook until lightly browned, about 3 minutes. Using a slotted spoon, transfer the mushrooms to a small bowl.
Add the potato, carrot, onion, garlic and butter to the skillet and cook until the vegetables are lightly browned, about 4 minutes. Stir in the flour. Pour in the wine and bring to a simmer, scraping up any browned bits from the bottom of the pan. Add the stock, thyme, bay leaf and chicken along with any accumulated juices and bring to a boil. Cover and simmer over moderately low heat until the potato and carrots are tender and the chicken drumsticks are cooked through, about 15 minutes. Discard the bay leaf and thyme. Stir in the heavy cream and the reserved mushrooms and season with salt and pepper. Sprinkle with the parsley and serve.