In a large Dutch oven or heavy bottomed pan over medium-high, heat the oil. Working in batches, add the lamb to the pan and sear on all sides, making sure not to overcrowd the pan. As each batch is browned, use a slotted spoon or tongs to transfer it to a plate.
When all the lamb has been seared, add it all back to the pan. Add the leeks, onions, carrots, parsnips, broth, bay leaves and thyme.
Bring the mixture to a simmer, then cover and cook for 1 hour.
After 1 hour, add the potatoes and continue to cook until the potatoes and the lamb are tender when pierced with a fork, about another 30 minutes.
Remove the bay leaves and thyme stems. Stir in the parsley and season with additional salt and black pepper as needed.