Cook potatoes in a saucepan in a small amount of salted water. (Do not overcook.)
Pour off all but 1/2 cup liquid.
To potatoes and liquid, add the clams, milk, and onion and celery soups; heat until simmering.
Remove from heat; add Cheddar cheese gradually, blending well after each addition. (If necessary, return to low heat to finish melting cheese.)
Do not boil.
Serve garnished with Parmesan cheese and parsley.