Parallel editing is when there are two different scenes that are being played together to give the effect that the two scenes are about the same plot.
A pioneer of parallel is D.W. Griffith as he was one of the first directors to use the parallel in the film the lonedale operator. In this film he used parallel editing to add suspense to a scene which cuts between three shots which are: a frightened girl, a robber trying to break in and the approaching posse.
A historic example of this is in the Godfather (1972) where on scene- the christening- is inter-cut (or cross cut) with a other scene- a series of murders. (from 3:35 till end of video)
I applied this technique to my first music video where the both of the main characters are getting changed in the dressing room (3:37-4:02) the parallel editing used in this scene makes it look like there getting changed together and at the same time however it is at two different locations and at two different times.