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In my personal JIRA Cloud instance, the parent issue keys are properly displayed in the sub-task view, for both Classic and Next-gen Projects:
That being said, please provide us the following information so we can better troubleshoot the problem:
1 - Is it happen for other sub-task issues too? Can you please try to create a new issue and sub-task and check if it reproduces?
2 - Can you try to use a different browser and check it in incognito mode?
3 - Are you able to properly access the parent issue of the sub-task? Or you are not aware of what is its parent? Maybe a security level is properly added to the parent issue, causing its child to don't display the parent issue key.
Let me know your results when you have the chance to test it.
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