What is the difference between parallel universe and multiverse?
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.