In a large pot, heat the olive oil over medium heat. Add the chopped onion and garlic, sautéing until the onion is translucent, about 5 minutes.
Add the diced carrots, celery, and potatoes to the pot. Cook for another 5 minutes, stirring occasionally.
Pour in the vegetable broth and bring the mixture to a boil. Reduce the heat and let it simmer for 15 minutes, or until the vegetables are tender.
Stir in the green beans, peas, asparagus, and dried thyme. Continue to simmer for an additional 5 minutes.
Using an immersion blender, carefully blend the soup until smooth. If you prefer a chunkier texture, blend only half of the soup.
Stir in the heavy cream and season with salt and pepper to taste. Heat through for another 2-3 minutes.
Serve hot, garnished with fresh parsley.