How to run drywall backer parallel to I-joist?
Yes, 2x2s attached to a joist will work as a nailer, use screws to secure the drywall there, although the drywall mounted to the wall will support it if it is tight to it. Another way to get nailers in place is to use short scraps of 2X material and nail them perpendicular to the joists, flush to the bottom of the joists at 16 or 24" on center and set your wall to them. That way you do not need to limit yourself to position your wall under a joist.