I'm trying to snap a vector control node to a grid so that it's perfectly parallel to the anchor point. However, when I click and drag the control node, the "snap point" seems to be relative to the place I initially clicked and not the actual control node itself.

Am I missing something obvious? How do I make an anchor handle perfectly horizontal/vertical?

Yeah, you encountered a bug here. I added a comment on this bug report about general snapping behavior: https://bugs.kde.org/show_bug.cgi?id=397559
I wouldn't know an easy solutions in Krita, at most you could insert temporary in-between points aligned to the grid and reset the handle positions with Segment to Line then Segment to Curve. https://docs.krita.org/en/reference_man ... ol-options
Maybe better to use Inkscape or other vector editors for the moment.