The ability to change the parent of a subtask is implemented as part of the Move permissioned workflow.  The Move Issue permission is defined as moving an issue from one project to another and is by default permissioned only at the project admin level.

the need to reparent a subtask is just a general user need within a project.  It should not be tied to this move permission.  As it is, general project users do not have access to this workflow that exists under the Move menu workflow on the subtask.

It would be better to have a more direct "Reparent" action under the action menu and not have this workflow under the Move action for subtasks.