We have subtasks that need to be executed in a very particular order. The subtasks have been ordered correctly when looking at the parent issue, by using the up and down arrows. This does not seem to actually update their ranking though.

When looking at the rapidboard view, the subtasks are not in the priority order they were put in under the parent issue. The board is ranking by rank. If I go back to the parent issue and use the gear to "rank to top" or "rank to bottom" then the order is updated on the rapidboard but not in the order on the parent issue.

The two are disconnected from one another. You would have to manually rank them twice, using each ranking UI individually. Meaning that you would first go in and rank them all in the Parent Issue view. And then rank all of the sub tasks again on the board.

I would expect the behavior to function basically the way updating ranking works when filtering by Epic on a backlog. You can update the relative order in just that epic. Then when you clear the filter the order of other issues has not changed. Meaning that if I update the ranking of subtasks on a parent issue, that is changing their relative order. So when looking at the Agile board, other issues may be sprinkled between them but the relative changes are reflected.