Sauté vegetables: 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 garlic and cook for 30 seconds.
Add peas and broth: Stir in split peas, potato (if using), broth, bay leaf, thyme, and smoked paprika. Bring to a boil, then reduce to a simmer.
Simmer: Cover and simmer for 45-60 minutes, stirring occasionally, until peas are tender. Add more broth or water if too thick.
Blend: Remove bay leaf. Use an immersion blender to blend partially for a creamy-chunky texture, or blend fully for smooth soup.
Season and serve: Taste and adjust with salt, pepper, and lemon juice. Garnish with parsley and serve hot.