Allow users to publish child pages of the pages they have access to edit

I have created a custom role called Custom Author. Users in this role can edit their own pages and add new pages. I would like to restrict new page creation and allow Custom Authors to create pages only as child pages of their own pages. In other words, I would like them to still be able to create new pages, but only as child pages of the pages they have editor access to.

How can I let users to access plugin functions based on roles?

I use a plugin called “Live Edit” to let authors and editors edit the content front-end. But the issue is that the plugin lets any logged in users to edit the content. I want the edit(specified by plugin. Not the WP’s default edit button) button be accessed based on their roles. I don’t know how to make it accessible based on roles. So trying to make it appear only for the authors and higher roles to that.