If using fresh clams, scrub and rinse them thoroughly. If using canned clams, drain and set aside.
In a large pot, heat the olive oil over medium heat. Add the diced onion, celery, carrot, and minced garlic. Cook until the vegetables are softened, about 5 minutes.
Add the diced potatoes, tomato paste, dried thyme, and bay leaf. Stir to coat the vegetables with the tomato paste.
Pour in the crushed tomatoes and chicken broth. Bring to a boil, then reduce the heat and simmer for 20 minutes, or until the potatoes are tender.
Add the clams to the pot and simmer for an additional 5 minutes, or until the clams have opened (discard any clams that do not open).
Stir in the heavy cream and season with salt and pepper to taste. Simmer for another 5 minutes to heat through.
Remove the bay leaf from the chowder before serving.
Ladle the tomato-based clam chowder into bowls and garnish with fresh parsley.