Heat the olive oil in a large pot over medium-high heat. Add the chopped onion and celery. Cook for about 3 minutes, stirring, until they begin to soften. Add the minced garlic and cook 1 minute more.
Pour in the chicken broth and water. Stir in the split peas, bay leaves, and thyme. Season lightly with salt and black pepper. Remember you can always add more salt later.
Nestle the ham bone into the pot. Bring everything to a boil, then lower the heat to medium-low. Cover and let it simmer gently for 60 to 80 minutes. Stir occasionally so the peas do not stick. The peas and ham should be tender.
Carefully remove the ham bone and set it aside to rest for about 10 minutes. Shred or dice the meat from the bone and cover it to keep warm.
While the ham rests, add the chopped carrots to the soup. Cover and continue to simmer for about 30 minutes more, stirring occasionally. The peas should mostly break down and the soup will thicken.
Stir the shredded ham back into the soup. Taste and add more salt and pepper if needed. Serve warm and sprinkle with chopped fresh parsley if you like.