A few people reported to me they couldn’t create an account in my app recently. After updating the Cloud Code to having more logging I saw the error.
This was happening in an afterSave hook on Parse.User, which was saving the user object again with an additional property. It seems the new Parse Session functionality (see http://blog.parse.com/announcements/announcing-enhanced-sessions/) has added this restriction in the cloud code.
The solution was to use the master key by calling Parse.Cloud.useMasterKey() before saving the user object.