With a malware infested P2P network, what you are working on is called a botnet.
A botnet is a collection of computers that are under the control of a person or group, usually facilitated by malware. These computers are often used to take part in DDoS attacks and spamming.
It's pretty much illegal to have a botnet in the vast majority of the world, but assuming you had one... yes, it's very possible. There have been cases where botnets (like several game "Anti Cheat" systems such as PunkBuster; anyone who is banned is forced to be actively ignored by every member of the botnet system, and they will relay it on to everyone else) have been used for massively expensive computations in parallel to do things such as mine bitcoins.
Parallel processing isn't just restricted to botnets, however. "Cloud Computing" is a form of parallel processing. As are programs for highly intensive research, requiring a supercomputer (such as folding@home). Without this, we wouldn't have MapReduce, the algorithm behind Google Search, either.
Like every piece of Anime or Manga, there'll always be something overlooked.
- Parallel processing is explainable with batteries in parallel, but overlook substantial amounts of detail such as concurrency and scalability.
- Limewire's known for turning your PC into a living colony of viruses. There's no "Single Virus" that created a botnet from it.
- "P2P" does not mean warez and torrent sites/programs, however much the media may be trying to push that. It's a system for distributed computing that just happens to be a great way to share data with low bandwidth.