What’s tripping you up here isn’t permissions or syntax, it’s how Time Machine behaves with network backups. On a Synology, Time Machine stores everything inside a sparsebundle, and tmutil can’t delete individual backups unless you point it at the mounted backup path plus a specific timestamp.
Running tmutil delete by itself or passing a snapshot name won’t work, which is why you’re seeing that error. The bigger issue though is that manually deleting backups inside a sparsebundle on a NAS often corrupts the backup set, especially on newer macOS versions, and Time Machine may silently stop trusting it.
The only safe ways to reclaim space on a NAS are letting Time Machine prune old backups on its own, increasing the quota on the Synology, or blowing away the sparsebundle and starting fresh. If you truly need to surgically remove old backups, you’d have to mount the sparsebundle read-write, identify the exact backup path under Backups.backupdb, and delete by timestamp, but that’s risky and not something I’d recommend unless you’re prepared to lose the entire backup history.