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1. To prepare the ribs, heat the oven to 350 degrees. If the butcher has not removed the membrane on the back of each rack, gently pry it up by sliding a sharp implement (like the tip of an instant-read thermometer) under it, then lifting gently. Grab the membrane with a paper towel and peel it off.
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2. Spread a 24-inch sheet of heavy-duty aluminum foil, shiny side up, on a work surface. Place one rack on top, rub it all over with oil, and generously season both sides with salt. Place 2 parsley sprigs and 2 garlic cloves under the concave side of the rack and 2 thyme sprigs on top. Wrap the ribs in the foil, pleating the edges to seal well. Repeat with the second rack. Place the rib packets in a large roasting pan.
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3. Roast the ribs for 30 minutes, then reduce the temperature to 250 degrees. Cook 1 1/2 to 2 hours more, until the meat has shrunk back from the ends of the bones by 1/4 to 1/2 inch and the ribs are tender enough to pull apart with your fingers.
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4. Meanwhile, prepare the barbecue sauce. Place the balsamic vinegar in a large nonreactive saucepan. Bring to a boil over medium heat and cook until reduced by a third. Add the remaining barbecue sauce ingredients with 1/4 cup water, bring back to a boil, then reduce the heat to low and simmer until thick, 30 to 40 minutes. If the sauce starts to thicken too much, add a little water. The sauce should be highly seasoned; adjust to taste by adding vinegar, brown sugar or salt.
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5. Remove the ribs from the oven and let cool briefly, then open the foil, being careful of the escaping steam. Transfer the ribs to a baking sheet. Turn on the broiler or raise the oven to 450 degrees.
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6. Slather the ribs on both sides with the barbecue sauce. Broil the ribs until the sauce sizzles and browns, 2 to 4 minutes on each side. Alternatively, bake in the oven 8 to 12 minutes. Baste with the barbecue sauce and serve at once with any remaining sauce on the side.
