There are no such things as parallel lines on a globe.

A "line" on a sphere is a Great Circle, and a "line segment" is an arc of a great circle. Any two distinct great circles intersect exactly twice, and so there are no parallel lines.

We call "lines of latitude" "parallels" but they aren't really, because they better thought of as curves than straight lines. They are "Equidistant curves" or "lesser circles" which correspond to ordinary circles in Euclidean Geometry.

To best plot an equidistant curve, my first thought would be Oblique Equidistant Cylindrical that is tangent along the fault.