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?
In parasitic interactions, one species (the parasite) benefits from the interaction while the other (the host) is harmed. So, whether an interaction is parasitic or not really depends on whether on the host species is harmed by the interaction. If something only grows on the dead remains of another organism, but causes no harm to living organisms in terms of growth and reproduction, it would not be a parasite, and might actually be beneficial to that organisms in some ways.
However, dead cells or tissues are not always useless to a still living organism. For example, the vast majority of cells in a mature tree are dead, but are still important for the survival of that tree by providing structure and water/nutrient transport. A hypothetical parasite could feed only on the dead cells inside of a tree and still cause harm to that tree by weakening it's trunk or impairing it's ability to transport water and other nutrients.
Not knowing much about fungal physiology myself, I can't say whether these &quot;hyperparasites&quot; are truly parasitic or not. But it is at least feasible for a parasite to only grow on the dead portions of a host and still cause it harm.