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Okay... so this draws on a couple of things which can be confusing. 1) perspective projections 2) mapping spheres onto 2D planes.
Usually when we think of a field of vision we imagine some projection that maps the 3D world in front of us to some 2D rectangle image. And that's all fine and well. We don't expect the lines in the image to conserve the angles they had in 3D.
I think what the author of the post is saying is that if you use a cylindrical projection that wraps around 360 degrees horizontally, then the lines will appear parallel when you unwrap it. But there's nothing wrong with this. If it seems like it would be a contradiction, because the lines cross each other at right angles in 3D - it's because in a z-aligned cylindrical projection, the point where the lines cross will be on one of the singularities that sit on each pole. And if the cylindrical projection is not z-aligned, the lines won't be parallel, and will cross each other at some angle.
I guess you can also think of this as two projections. There is the two lines on the floor, which are projected up onto the bird's panoramic view (a sphere), and then the sphere is projected onto a z-aligned cylinder, and then the cylinder is unwrapped to give us our 2D image with the two lines parallel.
Like how if you projected two perpendicular lines up onto the bottom of this globe they might align with say, 0"/180" and 90"/270", but they would appear parallel on the output cylindrical projection