Hi Ron,
This is not currently possible in Scrum boards. This is possible in Kanban boards though now. There is an existing feature request for this for Scrum boards in https://jira.atlassian.com/browse/JSWSERVER-7992
I would recommend watching this ticket for updates.
This has been discussed extensively in other threads though too. I would recommend checking out @Alex Kwiatkowski's work around on the thread https://community.atlassian.com/t5/Jira-Software-questions/Show-subtasks-on-Jira-Agile-Planning-Boards/qaq-p/455277
It is not exactly the same functionality as having the ability to manage subtasks on a backlog/sprint. However this method would at least allow you to show the subtask issue keys on each card of the parent issue.
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