I have some existing lines on my map, and I need to draw lines parallel to them at various distance. GRASS (GIS software I'm using) will draw those lines easily enough, but I want to be sure they're parallel on the globe, not just the map projection. What projection will make parallel lines on the globe parallel on the map? It doesn't need to be a whole-globe projection, I can split into sections if need be.

(Context: the lines I have are ocean ridges. I intend to create a set of lines parallel to them, and assign depths to those lines (ocean floor gets deeper away from the ridges; there's a formula that describes this quite precisely. Then I can interpolate a DEM from those depths.)