Give this a try:
Q) Why do I get the error "Permission denied (13) while creating "Project Cache..." when trying to create a Project?
A) This is typically seen on Mac systems where permissions are set incorrectly.
Get Info on the Pro Tools folder in Documents
Set Privilege it to Read & Write for your user account. (Check the Project Cache folder inside the Pro Tools folder to make sure the change was applied)
Here is a link to the
Pro Tools First FAQ page, where that was copied from.
Let us know if that fixed the issue for you.