Chop onion. Rinse mushrooms and trim discolored stem ends.
Put chicken in an 11- to 12-inch nonstick frying pan over high heat. Lightly brown pieces, turning as needed, 3 to 5 minutes. Remove chicken from pan and set aside.
Add onion, mushrooms, and 1/2 cup broth to pan. Stir to free browned bits, cover, and cook 5 minutes, then uncover and stir often until mushrooms are lightly browned, about 5 minutes more. Remove from heat and stir in vinegar, tomato paste, thyme, and remaining broth. Then return chicken and juices to the pan.
Meanwhile, with a sharp knife, cut off and discard top 1/3 of each artichoke and trim discolored ends of stems. Break off and discard leaves down to the tender, pale green inner ones. Cut artichokes in quarters vertically. With knife, cut fuzzy chokes from artichoke hearts and pull out the tiny leaves with thorny tips. Immerse artichoke quarters in about 2 quarts water mixed with the 2 tablespoons vinegar (so artichokes will darken less). Swish artichokes to rinse, then drain.
Lay artichokes on top of chicken, cover pan, and bring to a boil over high heat. Reduce heat and simmer until chicken and artichokes are tender when pierced, about 15 minutes.
Sprinkle parsley over chicken mixture, then spoon onto plates and season to taste with salt and pepper.