A "parallel dimension" means a separate and different reality that exists alongside our own, with its own set of laws and conditions. It's like a separate world that runs parallel to ours, but we can't interact or travel to it. Full definition
The new ability delivered to the gateway allows the characters to travel between parallel dimensions. (polygon.com)
It is based on a big storyline right now about parallel dimensions being destroyed in order to preserve the current version of earth. (goodereader.com)
There is some kind of story involving parallel dimensions and worm holes, which I suppose explains the crystals in some way. (deathbybeta.com)