Heat oil in a large pot over medium heat. Add onion, carrot, and celery; cook until softened, about 6 minutes. Stir in garlic; cook 30 seconds more. Stir in broth, split peas, thyme, bay leaf, salt, and pepper; bring to a boil. Reduce heat to low; simmer, partially covered, until peas are falling apart, about 1 1/4 hours.

Meanwhile, stir together parsley, sun-dried tomatoes, garlic, paprika, and salt in a small bowl.

Remove thyme sprigs and bay leaf from soup; discard. Add more broth or water to thin as desired, and more salt to taste. Serve topped with gremolata.