Rinse chicken, pat dry, and season with salt and pepper. Heat oil and 2 tbsp. butter in a large skillet over medium-high heat. Add half the chicken, skin side down, and brown, turning once, about 10 minutes per side. Remove and set aside. Repeat procees with remaining chicken. Remove from pan and pour off most of the fat, leaving just enough to thinly coat skillet.
Reduce heat to medium, add garlic and shallots, and cook, stirring frequently, until slightly soft, about 5 minutes. Deglaze skillet with vinegar and wine, scraping brown bits off bottom with a wooden spoon. Reduce vinegar mixture by about one-third, 3-5 minutes, then stir in tomato paste. Return chicken to skillet, pour in stock, and simmer for 10-15 minutes. Turn chicken and continue cooking until juices from chicken run clear, about 15 minutes. (If sauce becomes too thick, thin with a small amount of chicken stock or water.)
Remove chicken from skillet with a fork and set aside. Increase heat to medium-high, skim foam from sauce and continue cooking until sauce is thick and glossy, about 5 minutes. Cut remaining butter into small pieces. Remove skillet from heat and whisk butter into sauce. Adjust seasoning with salt and pepper. (Sauce should be smooth but tart; add additional vinegar if desired.) Return chicken to skillet, turning to coat evenly with sauce. To serve, transfer chicken and sauce to a serving platter and garnish with parsley.