Summary

Sub-tasks can be reordered within its Parent issue view by dragging and dropping the sub-task card.

The problem is that not all users with the same permission are able to reorder sub-tasks. Clicking on the sub-task card does not allow the reordering and only displays the issue information. 

Environment

Jira Cloud (With Subtasks enabled)

Diagnosis

Reordering of Sub-task does not need any additional permission, thus this is not an issue with permission. 

Cause

Sub-task list has a Filter to 'Hide done' sub-tasks from the view. 
The reordering in the child issues panel will be disabled when the “Hide done” filter is enabled. 

Solution

When you toggle the 'Hide Done' filter is on, it will remain persistent for all issues in that project.
The user that is unable to reorder sub-tasks will need to verify this and ensure to uncheck 'Hide Done' filter to be able to reorder the sub-tasks.