I take it that an important claim of secular Buddhism is that the Buddha never actually taught literal rebirth.
'Birth' ('jati') is the production of ideas or views of 'self' or 'beings' ('satta'), as explained in SN 12.2, SN 23.2 and SN 5.10. Therefore, the Buddha certainly taught literal 'rebirth' but this literal rebirth is the literal re-production of ideas & views of 'self' or 'beings' or 'persons'.
What happens during parinibbāna according to secular Buddhism?
Parinibbana is described in the suttas as the cravingless ending of conscious experience of an arahant (Iti 44; MN 140; SN 22.85; etc), as follows:
'Here in this very life, all that is experienced, not being delighted in, will be extinguished.'
How is it different from any other death?
'Death' ('marana') is the idea that a self or "a being" dies (SN 12.2). Parinibbana is not 'death'. There are many suttas that say an arahant does not experience death (SN 22.85; MN 140; MN 38; etc).
If there is no difference, how do secular Buddhists explain the very existence of this term?
Parinibbana is a term used for the peaceful cravingless ending of the life of an arahant. Currently, I do not know the reason why this term is used but I imagine it is possibly used so Arahants & Buddhas are not insulted by applying the word "death" to them and to highlight how the life of an arahant ends peacefully without any craving. The suttas say:
Dhp 21. Heedfulness is the path to the Deathless. Heedlessness is the path to death. The heedful die not. The heedless are as if dead
The Buddha-To-Be searched for freedom from "death" (MN 26) yet the question here does not focus on what "death" really is but wants to talk about Nibbana, which is something that will never ever be experienced if what "death" & the "deathless" really are is not discerned.
Can any death be called parinibbāna?
Death (marana) is not Parinibbana (cravingless ending of consciousness), as explained. Death is a personal self-view while consciousness is merely an impermanent impersonal natural element (dhatu).
"Death" is the death of beings or selves, as follows:
Long have you (repeatedly) experienced the death of a mother. The tears you have shed over the death of a mother while transmigrating &
wandering this long, long time—crying & weeping from being joined with
what is displeasing, being separated from what is pleasing—are greater
than the water in the four great oceans.
“Long have you (repeatedly) experienced the death of a father… the death of a brother… the death of a sister… the death of a son… the
death of a daughter… loss with regard to relatives… loss with regard
to wealth… loss with regard to disease.
When life & experience is viewed as merely elements & aggregates, there will be no more "mother", "father", "son", "daughter", etc, and no more "death".
And what may be said to be subject to death... sorrow... defilement? Spouses & children... men & women slaves... goats & sheep... fowl &
pigs... elephants, cattle, horses, & mares... gold & silver are
subject to death... sorrow... defilement. Subject to death...
sorrow... defilement are these acquisitions, and one who is tied to
them, infatuated with them, who has totally fallen for them, being
subject to birth, seeks what is likewise subject to aging...
illness... death... sorrow... defilement. This is ignoble search.