In a Dutch oven, cook bacon over medium heat until crisp. Remove bacon to paper towels, crumble, and reserve. Drain clams, reserve juice.
Cook and stir carrot, celery, and onion in bacon fat until onion is transparent. Stir in flour, cornstarch, potatoes, salt and pepper. Stir until all is coated with roux.
Add reserved clam liquor to jarred clam juice to make 3-1/2 cups; stir juice and milk into the vegetable mixture. Heat just to boiling, and reduce heat to simmer. Cook, stirring occasionally, until mixture thickens and potatoes are done.
Add clams and cheese, stir until cheese is melted. Sprinkle in reserved bacon. Garnish with copped parsley.