Season the short ribs with 1 tablespoon thyme and the cracked black pepper. Cover, and refrigerate overnight.
Take the short ribs out of the refrigerator an hour before cooking, allowing to come to room temperature. After 30 minutes, season them generously on all sides with salt.
Preheat the oven to 325 degrees F.
Heat a large Dutch oven over high heat for 3 minutes. Pour in 3 tablespoons olive oil, and wait a minute or two, until the pan is almost smoking. Place the short ribs in the pan, and sear until they are nicely browned on all meaty sides. Depending on the size of your pan, you might have to sear the meat in batches. Do not crowd the meat and don't hurry; this step it will take at least 15-20 minutes.
Turn the heat down to medium, and add the onion, carrot, celery, and thyme springs. Stir with a wooden spoon, scraping up all the crusty bits in the pan. Cook 6 to 8 minutes, until the vegetables just begin to caramelize. Add the balsamic vinegar and red wine. Turn the heat up to high, and reduce the liquid by half.
Add the stock and bring to a boil. Arrange ribs in the pot, lying flat, bones standing up, in one layer. The stock mixture should almost cover the ribs. Tuck the parsley sprigs in and around the meat. Cover tightly with aluminum foil and a tight-fitting lid if you have one. Braise in the oven for about 3 hours.
To check the meat for doneness, remove the lid and foil, being careful of the escaping steam, and piece a short rib with a paring knife. When the meat is done, it will yield easily to a knife. Taste a piece if you are not sure. [If you would like to cook these a day ahead, this is where you can pause. The next day, you can remove the fat easily from the pot -- it will have solidified at the top -- bring these back to a simmer on the stove or in an oven, and continue.]
Let the ribs rest 10 minutes in their juices, and then transfer them to a baking sheet.
Turn the oven up to 400 degrees F.
Place the short ribs in the oven for 10 to 15 minutes to brown.
Strain the broth into a saucepan, pressing down on the vegetables with a ladle to extract all the juices. Skim the fat from the sauce (if you made these the day before, you will have already skimmed them) and, if the broth seems thin, reduce it over medium-high heat to thicken slightly. Taste for seasoning.
Serve the short ribs with mashed potatoes and spoon some of the braising juices on top.