If you're receiving this error, it's because Parsec can't write the Parsec Authentication File to the default directory. There are permissions issues in the directory that Parsec was installed (locations for default installation are below). You can try to create a text file there and see if you can save it. If you cannot, your user needs permissions to this folder.
You can find Parsec in AppData (for Windows), or /home/<user> (macOS, Raspberry Pi, Linux), or Android's indeterminate extended data directory.
The other common causes for that would be insufficient privileges to create the file/dir, full disk, or unusual setup that doesn't allow files to be created in these normal places.
If you're on Windows, and you want a lower permission version of Parsec, you can download it here. Please note, this will not allow you to host on the PC you're using.