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.
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.
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.
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.
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.
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.