Heat oil in a Dutch oven over medium-high heat. Brown lamb well on all sides, in batches if necessary (this will take about 10 to 15 minutes). Remove from pan.
Add onions and garlic to pan and cook for a few minutes. Add wine, bring to a boil and cook for 5 to 8 minutes, or until liquid is reduced by about half.
Add tomatoes and thyme and bring to a boil, breaking up tomatoes with a spoon. Return shanks to pan. Place parchment paper directly on surface of lamb. Cover with lid and cook in a preheated 350°F (180°C) oven for 2 1/2 to 3 hours, or until meat is very tender.
Remove shanks from pan. Skim any fat from surface of sauce and discard. Puree sauce in a food processor or blender and return sauce to pan.
Remove lamb from bones in large chunks and return to wine and herb sauce. Heat thoroughly. Garnish with parsley, if using.