Not easily, the permissions are set at a project level, not issue type.
However, you can do it - if you have two separate workflows in the projects, one for Story and Epic, and another for bugs, you can set properties on the workflow steps that can allow/deny the delete permission. See https://confluence.atlassian.com/jira/workflow-properties-189949.html
On the other hand, there's a standard recommendation that you do not ever allow "delete", as delete really means it. The issue is destroyed and you have to look for backups if you make any mistakes. It's better to have a "deleted" status in your workflow, or a wastebin project to move them into.
I concur with Nic. You'll find tons of requests looking for how to restore deleted issues. There was even on earlier today. it WILL come back to bite you sooner or later. If nothing else than someone questioning why there is a gap in the issue numbers (JIRA doesn't reuse the number) and what was deleted and why.
Thank-you both for your answers. I appreciate your advice and opinions. We are thinking of allowing stories and epics to be deleted and understand that once they are deleted they are gone. So far I'm just investigating and looking at the options.
Do have an advice on what to do with tasks that no longer required for a story? Our users want to be able to use a template to create stories with a standard set of subtasks. They would then like to be able to delete the subtasks they no longer require.
Close them as "no longer required" is the standard approach for killing off the stories you don't want.
There is no "template" in JIRA for issue creation, and it doesn't natively create subtasks, you have to do it by hand or install add-ons. Instead of just creating and discarding, it would be better to adjust the creation process, so that when a story is created, it only creates the subtasks you actually want - e.g. one per component, or one set if they select X in a field or a second set if they put in Y.
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