Fill a pot or saucepan with 3 quarts water, and heat it to boiling.
Tear off any tough leaves at the base of the cauliflower and cut out the core; break or cut the large branches in to small florets, drop them into the boiling water, and cook until tender, about 5 minutes.
Lift from water, and drain well in a colander; when cool, put the florets in a large bowl.
As soon as the cauliflower is out of the pot, return the water to a boil and drop in the potatoes; cook them at a gentle boil just until a knife blade pierces the center easily--don't let them get mushy.
Drain and briefly cool the potatoes; peel them while still warm, cut into 1/4-inch dice, and put them in the serving bowl with the cauliflower.
Peel and core the apples; cut into 1/4-inch dice, and add to the salad bowl.
Trim the radishes, quarter them in wedges, and drop into the bowl along with the diced onion, cubes of Asiago, and toasted walnuts.
Sprinkle the salt over the salad pieces and toss; drizzle the olive oil and vinegar over the salad and toss; sprinkle on the parsley and toss once more, and serve.