Symmetries of Parallel Transported Lines
Consider two lines, r and r', that are parallel transports of each other along a third line, l. Consider now the geometric figure that is formed by the three lines and look for the symmetries of that geometric figure. What can you say about the lines r, and r'? Do they intersect? If so, where? Look at the plane, spheres, and hyperbolic planes.
We need to understand that straight lines lying in a plane that do not meet and is continued indefinitely in both directions, is an inhuman definition to parallel lines.
If a transversal cuts two lines at congruent angles, are the lines in fact parallel in the sense of not intersecting?
We need to look at the symmetries.