They don't have any duty as far as governing the non-profit, but as majority shareholder of the for-profit subsidiary, the non-profit would still have a fiduciary duty to the subsidiary's minority shareholders.
That decision does not dispute what you quoted at all. That case is not even about the first amendment, it is based on California’s state constitution. In fact your link makes the exact claim as the one you say is false.
> the federal constitution's First Amendment contains only a negative command to Congress to not abridge the freedom of speech