1. In a 5- to 6-quart covered pan over high heat, bring about 3 quarts water to a boil. Add linguine, stir to separate, and cook, uncovered, until barely tender to bite, about 9 minutes. Drain, reserving about 1 cup cooking water. Rinse and dry pan.
2. Meanwhile, cut off bases of radicchio heads; rinse and drain leaves. Cut leaves into 1/4-inch-wide strips.
3. Set the pan over medium-high heat. When hot, add olive oil, onion, and garlic. Stir often until onion begins to brown, 4 to 5 minutes. Add radicchio and vinegar; stir until radicchio is limp, 1 to 2 minutes. Add cream, linguine, and enough of the reserved cooking water to moisten as desired. Stir until pasta is hot, 2 to 3 minutes. Add cheese and salt and pepper to taste; mix gently.
4. Transfer linguine to a serving bowl or plates. Sprinkle with parsley. Offer more cheese to add to taste.