Rinse the split peas in a strainer, and then combine them with the stock and water in a large soup pot. Bring to a boil.
Add the ham bone, celery, 1 tablespoon of the parsley, and the tarragon. Reduce the heat to medium-low and simmer, partially covered, stirring occasionally, for 45 minutes.
Melt the butter in a saucepan over medium-low heat. Add the carrots, onion, and leek. Cook until the vegetables are wilted, 10 minutes. Add them to the soup pot, along with the spinach. Simmer, partially covered, 30 minutes.
Remove the soup from the heat. Remove the ham bone, and shred the meat from the bone, removing any excess fat. Return the meat to the soup.
Add the sherry, pepper, and remaining 2tablespoons parsley. Heat through, and serve immediately.