Heat the oil in a large dutch oven on high until it simmers. Pat lamb shanks dry with paper towels. Season meat with salt and pepper; brown on both sides, in batches, 4 minutes per side. Remote meat to a platter.
Reduce heat to low and cook garlic cloves until slightly soft, 10 minutes. Remove garlic to the plate with the meat. Increase heat to high, add the onion, carrot and celery to the pan and cook until soft and golden brown, 8 minutes. Add anchovies and cook for 1 minute. Add red wine and cook until almost completely reduced. Add chicken stock, tomatoes and thyme. Bring to a boil; season with salt and pepper. Return the meat, accumulated juices and the garlic cloves. Cover and transfer to the oven. Cook until meat is tender, 1-1/2 to 2 hours. Remove meat to a bowl; tent with foil.
Strain sauce through a mesh strainer and return liquid to the dutch oven. Cook over high heat until slightly reduced, 10 minutes. Season to taste with honey, salt, and pepper, and stir in the parsley.