Raise Stepdown Living Room
I would get 2x8 boards and rip them to 6.25. It's the simplest and cheapest solution, and probably also the strongest.

If you go parallel, then you have to make sure that the new joists are directly on top of the old joists - otherwise, they're only being supported by the subfloor. But if you put them perpendicular, you can put them anywhere you want, and I'm guessing that the whole setup will be stronger overall, since the load on each upper joist will be distributed among multiple lower joists. So perpendicular is the way I would go.