You could say that one is a subset of the other. That is, a "multiverse" is the total collection of all the "parallel universes" that can possibly exist.
Also, "parallel universe" isn't really an "official" scientific term, and isn't very well defined. It appears more in science fiction, and evokes images of evil twins with goatees.
The word "parallel" is problematic because it implies some kind of relationship between our universe and another. This might be meaningful for some "multiverse" theories, such as the many-worlds interpretation of QM, but loses meaning for other theories, like universes that exist outside of our cosmological horizon, which are causally disconnected, and may even have different physical constants than our own.
I'm going to offer an interpretation based on the fact that parallel universes and the multiverse arose in two different realms of physics.
I usually hear of parallel universes as a description of the many worlds interpretation of quantum mechanics. For each random, quantum-mechanical event, there is a universe for each possible outcome. That is, at each junction, a parallel universe is "forked".
The multiverse is associated with cosmological principles that make this crudely-stated extrapolation: if a vacuum fluctuation of some background caused the Big Bang in our universe, why could there not have been or still be further Big Bangs that produce other universes?
I think that the cosmological multiverse permits universes that have different laws of physics from those of our present universe, whereas the parallel universes are expected to have the same laws of physics.
This is a bit interpretive, so I stand to be correct by anyone more familiar with either of these terms.