If a "parasite" is growing on a dead part of the host or on a completely dead host, does it still count as a parasite?

What I mean is the case where the parasite does not contribute to the death of the host (tissue). Instead, the parasite may start growing after the host (tissue) is already dead.

I came up with this question when I was following the link in this answer and read the PDF about a fungus on a fungus on a fungus on a tree. In the PDF, the author first says that the Hypomyces was growing on the dead bracket mushroom, and then says again that Rhinotrichella globulifera was in turn growing on dead portions of the Hypomyces.

I'm not a mushroom expert and the author also does not mention whether the parasites caused the death of their respective hosts.

So suppose the parasites only started growing after the death of the host, do they really still count as parasites? My understanding is that a parasite should seek a host that is not dead and try to leech nutrients without killing the host too quickly (at least until the parasite is ready for the next stage of its lifecycle and no longer needs the host).