No, the simplest explanation is usually the best. Look at how Demonreach talks. It uses odd words for things. It would not sound right if Demonreach called it a "tree". Parasite is the sort of odd word Demonreach would use to refer to the tree.

And the distinction you refer to is hardly important. My body digests food and my parasites aid in the digestion process. Just two different ways of referring to a process at different levels.

Except that the "parasite" is specifically mentioned in connection with the circulation of blood, not of digestion. Remember the tree is associated with IV tubes going into Harry. But *something* still needs to keep his heart beating. I think that some fragment of Lash remaining behind is the most likely explanation.

Demonreach said this: "Mab gave you breath. Here provided nourishment. The Parasite maintained the flow of blood."

That Here is Demonreach, its referring to itself, the island and everything on it. The Parasite, a third entity, that Demonreach is referring to could only have come from 2 places, before Harry was killed or after the time that Mab got his body. And who else has a vested interest in keeping Harry alive that can do such a thing? And its not with digestion or circulation, Mab saved his body and fixed the damage, Demonreach provided the nutrients for the body with the tree roots and the Parasite (personally I think its Lash or Lasciel) maintained the body until Harry returned to it.

The other thing to keep in mind is the human body can not run on the same red blood cells for 6 months, they only last between 100-120 days since their maturation in the marrow before being recycled. Parasite also means something that lives off something else and for the most part not providing returns for what it took and the Tree was clearly giving something to Harry and not living off him.

It was also stated in past books that the Fallen are like parasite to their hosts and only Nicodemus and Anduriel have what can be called a symbiotic relationship with each other.