The closest you can get to this out of the box is an Epic burndown report. That will let you track the completion on all of the issues under an epic. Jira does not have a burndown for issues with sub-tasks.
Try using a Cumulative Flow Diagram with a custom quick filter. If your main board/filter includes the subtasks then you might be able to create a quick filter in the board settings for the report that allows you to select all of the subtasks under a user story.
EDIT: Here is a bit more help. Set up the quick filter on the board with the jql "parent = TEST-1234" where the issue key is the user story with the subtasks. When you apply that quick filter it will pull just the subtasks for that user story. You'll have to edit the filter each time you want a different user story, but at least it's a halfway decent workaround. It might even work for the psrint burndown report. Good luck, let me know if this works!
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