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Try the following: Connect to the synology nas using SSH and "ADMIN ACCOUNT" and issue the following commands:

sudo su (This command will ask por admin password)
cd /var/packages/Plex Media Server/scripts
./start-stop-status start

To verify it is running yo can issue the following command:

ps -fea | grep target/Plex\ Media

The output will be something like this:

ash-4.3# ps -fea | grep target/Plex\ Media
plex 14153 1 1 Nov02 ? 00:15:57 /var/packages/Plex Media Server/target/Plex Media Server
root 14762 13792 0 12:27 pts/4 00:00:00 grep target/Plex Media