>> Which folders need the permission?
It has to be the website's root. The reason for it is that upgrade packages may affect any file including the 'web.config' which is in the root
>> Would granting permission to the Worker Process good enough.?
>> Which folders need the permission? How do I do this using Plesk?
>> Or do I need to log into the server remotely and give it permission from there?
If you log in remotely to the server, find website's root and add every NTFS permission (except for "Full Control" - it is not a required one) to either "Everyone" or the user under which IIS is running working process, that will do it.
>> How do I give it write permission?
You probably know the answer to this one, but still: To grant ASP.NET access to a file, right-click the file in Explorer, choose "Properties" and select the Security tab. Click "Add" to add the appropriate user or group. Highlight
the ASP.NET account, and check the boxes for the desired access.
>> How do I do this using Plesk?
I'm not sure there're people that are familiar with Plesk here. I would try to search/ask the question on Stackoverflow/ServerFault