A star chart of a parallel universe
Paris McKenna from an alternate reality
A parallel universe is another name for an alternate reality. The terms "alternate universe" or "parallel universe" are sometimes also used to refer to alternate realities, however such usage is inaccurate; a universe is a distinct separate place, divided from other universes by higher dimensions (and may have its own alternate timelines), while an alternate reality is merely a different quantum facet of any one universe, with the same laws of physics, just different histories.
A parallel universe, also called a parallel reality or alternate universe, is a self-contained universe running concurrently with the primary universe. (VOY episodes: "Deadlock", "Flashback")
There are different types of parallel universes. Some are separate dimensional planes. Some of these planes are governed by completely different physical laws, while some duplicate the primary universe closely. Some parallel universes exist within subspace, while others exist as completely different quantum realities.
The idea of parallel worlds had been around since the 20th century. While working on with the experimental Spore drive in the mid-2250s, Captain Gabriel Lorca began seeing the possibility of exploring them. Later, his ship found itself in a hostile parallel universe, seemingly as the unexpected result of a jump, though this turned out to be the deliberate manipulation of Lorca, a resident of that parallel universe who had been posing as his primary universe counterpart. (Star Trek: DIS episodes: "Despite Yourself", "Vaulting Ambition")