Nowadays we come across with these two terms - parallel universes and multiverse. I wonder if this is the same thing or its two entirely different principles because the names are a bit confusing to me. Can anybody please shed some light on this?
I'd like to see the notion "parallel universe" restricted to universes similar to each other in some sense. This applies best to the many-worlds interpretation of quantum theory. Two universes in this sense can be almost identical, just different in a couple of quantum states.
Two such parallel universes could be interpreted (in the simplest case) as two elements of a Hilbert space. A Hilbert space defines a metric as a measure of distance between two elements (here applied to universes). This may be seen in some analogy to points (universes as analog) in an Euclidean plane (many-worlds multiverse as analog). Here the notion "parallel" can make some sense. At least, it's possible to compare the items in some way (by the distance between two points/universes).
Talking of universes as parallel in the very general case of items in a multiverse, when there is no similarity at all between these items, would be a bit too lax in my eyes.
In my own mind the multiverse could consist of either multiple universes occupying the same "space" (as in different dimensions) and/or different areas of space where what we consider to be our own universe is only one "bubble", outside of which are an infinite number of other bubbles, possibly created by their own Big Bangs.
The first variant above is what I would refer to as parallel universes, i.e. not the "bubbles". Or to put it another way, The different universe instances in the multiverse could be parallel.
Note that my understanding of current hypothesis and theories in this area is very limited. These are just my own thoughts.