Are there lower bounds to complexity for hosts and how are these described or modeled? In other words, one would expect an organism to have a certain amount of "surplus tissue" in order for parasites to be able to feed on it, at least without killing the host too soon for the parasites to thrive. For example, humans have surplus blood (we can draw a lot of blood and survive), making it possible to live with a few leeches, ticks, etc. Is this supposition really true? That is, do we know that "very simple" organisms cannot have parasites? If so, how do we characterize what is "too simple to be a host" and does it have anything to do with having "surplus tissue"? (I assume "surplus" is not the term used in parasitology because I haven't seen it in the books...)
"Complexity" is a difficult term to operate with in biology; most people that use it don't have an idea of how they'd like to operationally define it, and without an operational definition it isn't meaningful.
Phages are effectively parasites of bacteria - one could bicker that a virus can't be a parasite because it's not "alive", but I think this is just a semantic argument about what it means to be alive, and many of the definitions that exclude viruses are problematic because they those same terms exclude animal parasites.
Not all parasites are of the "feed on the host just a little bit" variety, either. Parasitoid wasps, for example, typically consume the entire target host, sometimes paralyzing it at the same time an egg is laid. Cordyceps is another fun group of (mostly) parasites, whose targets include other fungi and insects.
The only limit seems to be that the host must somehow be of use to the parasite. It must also be complex enough to be considered a living organism in its own right: you can't have a parasite of something that's already dead, for example, since that has a different meaning. I think these meanings are rather tautological, rather than interesting, though, so I'd answer your title question to say "no, not in any meaningful, non-tautological way".