Pour the split peas onto a plate or into a shallow bowl and pick through them, removing any small stones, grit, or shriveled peas. Transfer to a fine-mesh strainer and rinse under cold running water until the water runs clear, then set aside.
Heat the olive oil in a large Dutch oven or heavy pot over medium heat. Add the diced onion, carrots, and celery and cook, stirring occasionally, for 6 to 8 minutes until the onion is soft and translucent. Add the minced garlic and cook for about 1 minute more until fragrant, being careful not to let it brown.
Add the rinsed split peas, bay leaves, dried thyme, and black pepper to the pot and stir to coat everything in the aromatics. Nestle the ham bone or ham hocks into the center, then pour in the chicken broth and stir to combine.
Increase the heat and bring the soup to a boil, then reduce the heat to low, cover partially with a lid, and simmer gently for 1 hour to 1 hour 30 minutes, stirring every 20 minutes or so and scraping the bottom, until the peas have broken down and the soup is thick.
Using tongs, lift the ham bone or hocks out onto a cutting board and let them cool for a few minutes. Pull the meat from the bone, discard the bone, fat, and any skin, then chop or shred the meat into bite-sized pieces and return it to the pot along with the diced cooked ham.
Fish out and discard the bay leaves. For a smoother soup, blend about half the pot with an immersion blender or mash some of the peas against the side with a spoon. Taste and season with salt as needed, thin with more broth or water if desired, then simmer another 5 minutes to heat the added ham through. Ladle into bowls and top with fresh parsley.