Edit access to attached files is down to whether a person can attach a (new copy of) a file or not. The space permissions allow you to say who can or cannot attach files, so you'll need to check that.
I'm afraid it does not go down to a page level unless you are willing to get apps that add new functions for more file access control. (I'm not that familiar with any of them, so I'm not sure what to look for in the marketplace)
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