Put lamb and cold water in a stockpot and bring slowly to a simmer. Skim well, then add salt, pepper, thyme, and bay leaves.
Simmer gently for 75 to 90 minutes until the lamb is tender. Remove the lamb, discard bones and excess fat, and tear the meat into irregular pieces. Strain the broth if desired.
Finish the Cawl:
Return broth and lamb to the pot. Add swede, carrots, and parsnips and simmer 20 minutes. Add potatoes for 15 minutes, then leeks for the final 12 to 15 minutes.
Discard thyme stems and bay leaves. Stir in parsley, adjust seasoning, and let the cawl rest 10 minutes before serving.
Notes
Cawl is often even better after a night in the refrigerator; lift off any firm fat before reheating.
Serve with crusty bread and a piece of Caerphilly or mature cheddar cheese.