Parinibbāna in secular Buddhism I take it that an important claim of secular Buddhism is that the Buddha never actually taught literal rebirth.

What happens during parinibbāna according to secular Buddhism? How is it different from any other death? If there is no difference, how do secular Buddhists explain the very existence of this term? Can any death be called parinibbāna?
In my understanding, the notion of Nirvana was a nod towards pre-Buddhist (Inc. Jain) ideas of the times. Nirvana was something transcental, something ultimate everyone spoke about but barely anyone could define.

When The Buddha achieved his Breakthrough, he mapped his ideas back to preexisting notions, to make it easier to communicate with people. So Nirvana is a bit tongue-in-cheek term, the so-called Nirvana. The Buddha attained what he himself called "Liberation of mind from suffering through Wisdom", which he equated with the Nirvana everyone at the times were looking for. 

Then, the notion of Parinirvana is an even more ironic gesture. Since "Nirvana in this life" was actualized by not holding on to anything that could create a basis for suffering, death of "someone who is like this" (Tathagata) was the ultimate act of not holding on to anything to the highest degree possible, thus it can be half-jokingly called Super-Nirvana or Parinirvana.

Regular people die while still grasping at things, and so their death leaves an element of regret which then lives on as a negative impulse affecting the lives of people to come.

The Buddha's death is the ultimate act of letting go. It's an act of letting things go on their own course, governed only by Natural Law. The Buddha haven't attempted to stay behind and continue teaching Dharma indefinitely - which would be an act of grasping. Instead, his Parinirvana is an example to all of us Buddhists to not force the right Dharma on everyone, nor be aggressive in teaching and saving people, but to let things play out naturally while helping whenever we can.