I got this error when enabling CloudFlare for one of the sites in cPanel control panel.

You must have the zoneedit feature to take this action. You currently only have the simplezoneedit feature.
THis issue was solved when i enabled "" from WHM (WebHostingManager - server manager, this is control panel of the whole hosting server, shared hosting accounts dont have access to it, only hosting provider).

Papckages > Feature Manager > Default package [Edit] > tick "Advanced DNS Zone Editor" and save