Normally Plex Media Server does not need write access to the media folders for the various libraries but for DVR Recording (and also if creating optimized versions), Plex Media Server would need to have full write access.
I would suggest checking the file and directory permissions for the paths defined for TV Shows library
What you could do is have a separate path for recordings and have that with write access but normal path just read
To find the path open Plex Web and click on the ... next to the library name and select Edit Library and then Add Folders
Then in linux shell session on the NAS check the permissions using ls command eg ls -al