Pre-Made Cables & Headers for (Parallel Interface) GLCD's
A place I used to work at connected a graphic LCD to the main board with what looked like a bunch of tinned strips sandwiched between Tyvek-type (non-woven polymer cloth) material.  

Nevertheless, is something like this (in this category, but not FFCs) what you're looking for?  Digi-Key doesn't seem to have 20-pin cables in-stock. Manufacturer (Molex) link.

Edit:
If you want to go the rigid route, headers come in all shapes and sizes.  Ones explicitly for this application I've heard them called double-level, mezzanine, but Digi-Key has them filed under board spacers/stackers, though of the 4900 they "carry", 4 are in-stock.

For prototyping male/female 0.100" headers can be nice.