|Jira Agile 6.3
When moving any or all sub-tasks to "In Progress", its parent issue remains in "Open" state. It does not reflect the current status of the sub-task, and the job at hand. The only time a parent transitions to a new status is when all of its sub-tasks are "Resolved" or "Closed". This is the default setting for JIRA Agile.
Ensure that you have the Misc Workflow Extensions plugin installed in your instance.
For OnDemand Users
The plugin has already been bundled but requires activation via Add-Ons > Manage Add-ons
Now, attempt to transition issues from one column to the another column in Work Mode. As long as you're transitioning a sub-task from a list of sub-tasks, you will notice that the parent issue's status will follow the moved sub-task's status: