Well yes, I understand Time’s leftwing agenda and biases. But Flickr is a big boy, with gazzilions more users than that fading rag.
Agreed. But the zero-effort way out is to simply respond to Time, delete a page, move along.
It's like "zero tolerance" policies are just an excuse for principals to not have to make difficult common sense choices.
Isn’t parody exempted?
Yes. For original works of Authorship.
If the photoshopper had started from his own Obama image, or even a public Whitehouse image, and even if they'd made it look like a Time cover as a parody, then he'd own the copyright[note].
As a derivative work, though, he has to have approval from the original content.
[note] Keep in mind though, TIME is in the mood to bluff, so they could still threaten them on that, and probably successfully again, even though the parody should win in court.