science fiction genre: universe (Q47306921) coexisting with another universe (such as Q1)
They're not the same thing. Taken from the definition of parallel universe:
A parallel universe or alternative reality is a hypothetical self-contained separate reality coexisting with one's own
And from the definition of alternate universe:
An alternative universe (also known as alternate universe or alternate reality), commonly abbreviated as AU, is a type or form of in which canonical facts of setting or characterization in the universe being explored or written about are deliberately changed.
So looking at the definitions, we see some differences. A parallel universe would be a completely separate universe, possibly containing similar characters or facts, but definitively a separate entity. An alternative universe would likely take place in the same universe, but with altered facts (i.e. "what-if" scenarios).
Using Star Wars as an example, as that's what I'm most familiar with, E.T. would be (mostly) a parallel universe to Star Wars. An alternate universe would be something appearing under the Star Wars: Infinities label (deliberately non-canon stories) or fan fiction not approved by Lucas Enterprises.
Writers aren't very strict when it comes to these two particular terms. Most of the time, the two terms are interchangeable, unless they've been defined to be different within that continuity.
For example, in Star Trek, the word "parallel" is taken to mean "alongside", not that the two universes developed in parallel/mostly in parallel. They have timeline splits for universes that were identical up until the divergence point, an antimatter universe that appears to be otherwise completely identical to ours, and at least one "parallel universe" that's on a difference plane of existence.
Does one exist within the other? Are they separate?