In a Dutch oven bring the water and 3/4 teaspoon of the salt to boiling. Meanwhile, place parsnips in a food processor. Cover and process until chopped; transfer to a large bowl. Repeat with cauliflower, then with carrots and onion. Add chopped vegetables to the boiling water; reduce heat. Simmer, covered, for 9 to 11 minutes or until vegetables are tender, stirring occasionally. Drain in a colander.
Return vegetables to the hot pan over low heat. Mash with a potato masher until vegetables reach desired consistency. Cook and stir over low heat for 2 to 3 minutes or until any excess water is evaporated. Stir in chives, parsley, cheese, butter, pepper, and the remaining 1/4 teaspoon salt.