Sprinkle the chicken pieces with salt and pepper. Melt the butter and olive oil in a saute pan and brown the chicken, a few pieces at a time. You're not cooking the chicken here, just making the skin crisp and giving it color and flavor. Five minutes per side is about right. Remove the chicken to a dish.
Add the garlic and cook for 5 minutes. Deglaze the pan with the vinegar and boil down by half, about 10 minutes. Return the chicken to the pan, and pour in the stock. Add the tomato paste, tomatoes, bay leaf and thyme. Simmer, uncovered, until the chicken is cooked, about 30 minutes. Remove the chicken to a clean dish and keep warm.
Strain the cooking liquid into a saucepan, pressing to get all the juices through, and whisk in the last spoonful of butter. Pour over the chicken. Sprinkle with the parsley and serve.