In math, parallel means two lines that never intersect — think of an equal sign. Figuratively, parallel means similar, or happening at the same time. A story might describe the parallel lives of three close friends.
Parallel is from Greek parallēlos, from the prefix para-, "beside," plus allēlōn, "of one another," from allos, "other." As a noun, a parallel is a way in which things resemble each other — you might draw parallels between the Vietnam War and the U.S. invasion of Iraq. In specialized use, a parallel can mean one of the imaginary circles on the surface of the Earth that are parallel to the equator — the 49th parallel divides the U.S. and Canada.