My Plex share to my parents keeps becoming unavailable

You are currently using UPnP to create the necessary port forwarding in your router.

Switch to the ‘manual’ method.
https://support.plex.tv/hc/en-us/articles/200931138-Troubleshooting-Remote-Access

Those automatically created port forwardings tend to get erased by the router after a while.

Do also disable power saving on the ethernet adapter on your plex server machine: