The Lotka–Volterra equations describe how predator–prey interactions affect population growth. Do these equations describe parasite-host interactions? If not, how would they change by adding these interactions as well?
Parasite vs predators
In terms of ecological interaction, a parasite is essentially a predator. As such, of course the Lotka-Volterra equations apply to host-parasites interactions.
However, Lotka-Volterra equations are very simple and very general. So much so, that in order to make good prediction about the real world, one most often have to build more advanced models.
Host-parasite interaction models
There exist a whole set of models specific to host-parasite interactions. You should have a look at the wikipedia entry for epidemic model which is a pretty good overview and introduction.
Most models of host-parasite interactions fall into the category of the SIR (or SIRS and other derivatives) model. The term SIR comes from Susceptible-Infected-Recovered. The whole point of the SIR models is to categorize host individuals on whether they are Susceptible, Infected of Recovered (hence the abbreviation SIR).
The SIR models are very much used by national and international centers for epidemiology and underly many of the decisions of vaccination and other methods of disease control.
Of course, the Lotka-Volterra model and the SIR models assume absence of evolution and only track changes in population size (within each category for the SIR models). There are models that also take into account that populations also evolve in response to the interaction and demographic events that the interactions are causing. Such models are much more complicated to keep track of and we often (I think) make use of technics such as separation of time scales. These models are essential though typically for to control viral epidemic but not only. But these models are outside the scope of your question.