In a large roasting pan or 2-gallon resealable plastic bag, combine the wine, port, garlic, celery, carrots, onion, thyme and bay leaf. Add the beef shanks and turn to coat with the marinade. Refrigerate for 24 hours.
Preheat the oven to 300°. Remove the shanks from the marinade and pat dry. Strain the marinade into a bowl, reserving the vegetables and marinating liquid.
In each of 2 large skillets, heat 1 tablespoon of the oil until shimmering. Season the shanks with salt and pepper and add 4 to each skillet. Cook over high heat until browned, 3 minutes per side. Transfer the shanks to a large roasting pan and arrange in a single layer.
Add the remaining 2 tablespoons of oil to the skillets. Divide the vegetables between the skillets and cook over moderate heat, stirring, until starting to brown, about 4 minutes. Stir 1 tablespoon of the tomato paste into each skillet and cook for 1 minute. Add 3 cups of the broth to each skillet and bring to a boil, scraping up the browned bits on the bottom. Pour the vegetables, broth and reserved marinade over the shanks and cover the roasting pan with foil.
Braise the shanks for 2 1/2 hours, until just tender. Uncover and braise for 30 minutes longer, until the shanks are glazed on top. Using a spatula, transfer the shanks to a large rimmed baking sheet; cover with foil.
Strain the braising liquid through a coarse sieve set over a large saucepan, pushing the vegetables through as much as possible. Skim the fat off the sauce and boil the sauce over high heat until reduced to 4 cups, about 35 minutes. Season with salt and pepper.
Return the shanks to the roasting pan and pour the sauce over and around them. Cover with foil and simmer over moderate heat for a few minutes to rewarm the shanks. Transfer the shanks to serving plates, spoon the sauce on top, garnish with parsley and serve.
The braised shanks can be prepared through Step 6 and refrigerated for 3 days. Skim off any fat, return to room temperature, cover and reheat in a 325° oven for about 25 minutes or until heated through.
This formidable dish needs a red wine with some tannic oomph, like a Napa Cabernet.