Is it possible for a user to have permissions set so the user can create sub-tasks but not parent tasks?
No, there's no way to do this directly, the permission to create an issue allows you to create an issue of any type in the project.
There is a clumsy trick - if you can create a validator that can check the user against the issue type, then it could fail the "create" transition and tell the user that they're not allowed to create issues of specific types. It's not too nice for the users because they lose anything they've put into the issue already.
UPDATE: It's a wrap for our first community AMA on Jira Work Management. Thanks to everyone for taking part and sharing. Feel free to keep the discussion going! Share your questions now Hi...