Pat chicken dry and season with salt and pepper. Heat oil in a large dutch oven over medium high heat until just smoking. Brown half of the chicken on all sides. Transfer chicken to plate and brown remaining chicken. Discard any fat from pot.
Add butter to pot and heat over medium until melted. Add onion, mushrooms, and 1/4 teaspoon salt. Cook, stirring occasionally, until light brown, about 10 minutes. Add garlic and thyme and cook for 30 seconds. Stir in flour and cook 1 minute. Stir in wine and scrape any browned bits from the bottom of pan. Next, stir in broth and bay leaves.
Nestle chicken and any accumulated juices into the pot and bring to a simmer. Cover, turn heat to medium low, and simmer until chicken is cooked and tender, about 20 minutes for breasts, or 1 hour for thighs and drumsticks. If using both types of meat, simmer thighs and drumsticks for 40 minutes before adding breasts to pot.
Transfer chicken to a serving dish and tent with foil. Skim as much fat as possible off the surface of the sauce. Stir in cream and return to a simmer. Cook until thickened and reduced to about 1 1/2 cups, about 15 to 20 minutes. Remove and discard bay leaves. Stir in lemon juice and season with salt and pepper to taste. Spoon sauce over chicken and sprinkle with fresh parsley.