That is what the Bishops would like for you to believe, but in most instances that is not the case. Most of the established parishes hold legal title to their property. Recent rulings in Calfornia dealing with Methodists and Presbyterians have held that legal title, absent an express trust of record, controls. So in California, at least, the local parish might well be able to hold their property. Outside California, you would need to research state law, which might still not be conclusive on the subject.
If the court is willing to use neutral principles of law, the parish is more likely to win; if the court defers to the religious body, the bishop is going to win.
Some of the parishes have elected to walk away from the property, even if there is a good likelihood of success, because they don't feel that the legal battle is a good use of resources.
Every state has different laws, so cases must be evaluated on a state by state basis. In Florida, the Diocese is listed as the legal owner of the properties.