What are these two nerves running parallel to the spine called?
From this closeup image, you can see there is more than one nerve on each side that is parallel to the spine.



Source: imgur.com

First, they are the right and left sympathetic trunk that run from the base of the skull to the coccyx.



Image source: Earthslab.com

Additionally, they could be the right and left long thoracic nerve, which innervate the serratus anterior muscles in the chest wall. In your image they appear like running along the spine, but in reality they run vertically and laterally in the chest wall.



Source: Semantic Scholar