How do predator prey mutualism and parasitism maintain balance within an ecosystem?

The simple answer is - they don't. The "balance" to which you refer is just a matter of where things settle out. While the various species are "balancing", some may die out in the environment, some may thrive, some may overpopulate to a detrimental level. The point at which the population levels of the various species stops changing is basically the balance point. It all depends on the resources available and the fitness of each species for obtaining the resources it needs to survive and reproduce. If conditions change, during a drought for example, then the species are going to go through that balancing process again. Some might die out. Heck, they might ALL die out. Different species might thrive. In other words, there is no guarantee that nature will support a given balance of species within a certain environment. Predator/prey, mutualism, and parasitism (and commensalism, etc.) are strategies species use to try to make themselves more fit and able to survive and reproduce.

Depending on conditions, some of these strategies are more beneficial than others.