I have an update re: this feature request since it has been almost a year since our last update. We are developing the ability to drag and drop subtasks, however, this is being released to customers as one piece of the new issue detail view for Jira Cloud. See 'A sneak peak into the future of Jira Cloud issues' for more details.
This is a lot more work overall but aims to address many different pieces of feedback at once, including this one. You may not have this enabled yet, as we are still working on some of the more complex aspects of Jira, but you can expect to have this enabled with the option to use it soon as we continue our rollout.
Lastly, there are many customers using the new issue view today. We are getting a lot of positive feedback but also some great feedback on areas we need to improve. We are investing in this new experience rather than fixing things on the old view. I have attached a small clip in the attachments section to show what the drag and drop for subtask functionality looks like.
Thank you for your patience and continuing to provide feedback. Please remember that jira.atlassian.com is one of many inputs for the JIRA roadmap. You can learn more about our process here.
tpechacek (at) atlassian (dot) com
Senior Product Manager, JIRA