Origins
Two lines that run beside each other forever, never meeting — that image is built into the word parallel itself. It comes from Greek parállēlos, composed of pará ('beside') and allḗlōn ('of one another'), a reduplicated form of állos ('other').
The word entered English in the 1540s through Latin, initially as a term of geometry. Euclid's parallel postulate — the idea that through a point not on a line, exactly one parallel line can be drawn — had fascinated mathematicians for two thousand years before the English word existed.
The figurative extension came quickly. By the 17th century, writers spoke of parallel lives, parallel events, parallel arguments. Plutarch's Parallel Lives, comparing Greek and Roman figures, gave the metaphor its most famous expression, though the English title came centuries after the original Greek.
Latin Roots
The Greek root állos ('other') spread widely through English by different routes. Alien comes from Latin alius ('other') — someone from another place. Alibi means 'elsewhere' — proof of being in another location. Allegory speaks of 'another thing' through symbolism. Allergy is a reaction to what is 'other' to the body.
Parallel lines are, at their root, simply others that travel beside you — close, constant, but never converging.