@rlopez Thank you for your reply above and your email (I'll reply to that as well)
Joe Riel's hint to consider the cone tangent to the unit sphere at the path of transport lit a light for me. I imagined the following:
On a rectangular sheet of paper (i.e. a 2d Euclidean space) draw successive images of a vector parallel transported along a horizontal line. Then draw on the paper the shape which will form the tangent cone. While still flat, translate the vector images without changing their orientation, from the roots on the horizontal line to roots on the arc which will be the line on the cone tangent to the unit sphere.
The translated vector images will appear to rotate horizonally vis-a-vis the cone's tangent line, as they will when the cone is formed from the paper cutout and placed on the unit sphere.
In this sense, the vectors on the unit sphere maintain the parallelism they had before Euclidean space was transformed into the tangent cone.
Does this sound correct?