Heat olive oil and melt butter in a large pot or dutch oven set over medium-high heat.
Add chopped celery, carrots, and onions; cook for 4 minutes.
Stir in garlic and cook for 1 minute.
Add chicken broth and water.
Stir in split peas, bay leaves, thyme, and oregano; season with salt and pepper. Don’t use too much salt because the ham hocks are already salty.
Add ham hocks to the pot and bring mixture to a boil.
Reduce heat to low, cover the pot, and let simmer for about 1 hour and 15 minutes. Stir occasionally. The soup is ready when the peas are soft. Start checking it for doneness around the 55-minute mark.
Remove ham from soup and let rest for several minutes, or until cooled enough to handle. Then, shred or chop up the meat into smaller pieces. Discard bones and skin.
Add chopped ham back into the soup. Cook for a minute.
Remove from heat.
Taste soup for salt and pepper; adjust accordingly.
Ladle soup into bowls.
Garnish with parsley and serve.