When you change your Plex account password, it unclaims your server, e.g. removes it from your account. You need to re-claim the server.
If you have remote access to the Synology NAS (Synology DSM, not Plex), then this should work.
If running DSM 7, see FAQ 31: Reset / Reclaim PMS on DSM 7 (Not Authorized - Locked Out)
If running DSM 6, see FAQ 15: Claiming & Connecting to a troublesome Synology server. Use DSM 6 Alternate Method 2.
