A technique protects user data within a storage system. The technique involves, in response to a request signal that directs the storage system to store the user data, generating first parity and second parity based on the user data. The technique further involves storing the first parity and the user data in a data uber of the storage system, the data uber providing redundant array of independent disks (RAID) protection. The technique further involves storing the second parity in designated free storage of the storage system, the designated free storage being separate from the data uber.