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 US 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 US and Canada.