Heat olive oil in a large pot over medium heat. Add onion, carrots, and celery, and sauté for 5-7 minutes until softened.
Stir in minced garlic and cook for 30 seconds until fragrant.
If using a ham hock, nestle it into the pot. If using bacon, cook until crispy, remove half for garnish, and leave the rest in the pot.
Add split peas, broth, bay leaf, and thyme. Bring to a boil, then reduce to a simmer.
Cover and simmer for 45-60 minutes, stirring occasionally, until peas are soft and soup thickens. Add potato halfway through, if using.
Remove ham hock, shred meat, and return to pot. Discard bay leaf. Adjust seasoning with salt and pepper. Serve hot, garnished with parsley or bacon bits.