As mentioned, activating on the new computer which runs the same OS doesn't work regardless. It also gives a message: "Unable to sign in. Please check your internet connection". My UID and password are correct and no firewall is enabled. So the steps as described in that KB article cannot be...
I am having the same issue. I am trying to sign-out of Parallels Desktop 11 to de-activate it on one computer and activate it again on another. Sign-out on the old one does not work ("Unable to sign out. Please check your internet connection"). The new computer does recognise the license key...