I'm sorry, but this is a very broken process you have defined here, and hence it's not going to be possible for you to get even close to it in Jira.
Taking it a piece at a time:
> I have a defect issue type that is created as a sub task for a Bug Issue type.
Jira does not limit sub-task types by the parent
> I have another custom issue type called 'triaged defects' . Any unresolved or triaged defects that will not be closed in the current sprint will be moved to the triaged defects custom issue type.
You should never have "change issue type" as part of a process. Changing issue type is for housekeeping and when someone has made a genuine mistake. Because of this, it is a nightmare to do it in Jira - it simply isn't built to do it.
> I would like to restrict permissions for the defect subtask to be able to move under other issue types such as story , epic or Task.
Nope, one of the things you can't do is conditionally protect "move". You can either move an issue or you can not.
> For the defect sub task , when I select the change parent
This is also something you should not be doing as part of a standard process. It is housekeeping or error, but also something you might do sometimes as part of backlog refinement, but still not a standard process.
> How do I achieve this using script runner.
With an enormous amount of very complex code that everyone I've seen attempt it gives up on.
I very strongly recommend you take a step back, look again at what you are trying to achieve and redesign the process so that you do not need to change issue types or parent issues as part of it.
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