No, parallel lines are straight lines.
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The Missouri Compromise (1820) settled on the parallel 36 degrees 30 minutes. North of that line, there could be no new slave-states.
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An invisible line that forms east-west circles around the Earth is called both parallel and line of latitude. An imaginary line that circles the earth from north to south is called both meridian and line of longitude.
The territories acquired from France in the Louisiana Purchase of 1803, which were awaiting admission to the USA. The Compromise put down down a line in the sand (parallel 36,30), North of which slavery would be illegal.
It was the parallel that marked the Southern border of Missouri. All the way to the Pacific, anywhere North of that line was free soil. This was meant to settle the slavery argument in respect of new states. Both sides were equally dissatisfied with this solution - but 'equally' was the point. It kept the balance, and it kept the peace for thirty years.