Combine split peas, water, celery, parsley, tarragon and bay leaf in a large pot or soup kettle over high heat and bring to a boil. Reduce heat and simmer briskly, stirring occasionally, until split peas have become quite soft and start to dissolve into the liquid, about 30 minutes.
Meanwhile, heat olive oil in a large skillet over medium heat until shimmering. Add carrots and leeks and cook, stirring occasionally, until softened and just beginning to brown, about 10 minutes.
Add carrots and leeks to soup, along with generous amounts of salt and pepper. Reduce heat and simmer briskly until soup thickens, about 30 minutes.
Remove and discard bay leaf, season with salt and pepper to taste, ladle soup into individual bowls and garnish with minced parsley.