This might depend on how you have sharing set up, and what access permissions you might have on the destination folder on the Mac side that you tried to copy to.
Test things out by first trying to copy your Thunderbird profile folder to somewhere else within your virtual Windows setup, such as to your Windows desktop.
* If that generates the "access forbidden" error, then you know that something is weird on the Windows side.
* If that works just fine, then it sounds like you need to look at your sharing setup and your folder access permissions on the Mac side. To check your sharing setup, in Parallels, click Virtual Machine -> Configure, then select Sharing in the list on the left. To check your Mac folder access permissions, on the Mac side, select the folder in question and click File -> Get Info, then look at Sharing & Permissions at the bottom of the info dialog.
Hope this helps,