Can a joint custody parent give the school permission to send and receive email about a child to his girlfriend without the other parents permission?

No. That would be a violation of the child's and the parent's privacy. The girl friend has no right to receive any information about the child. The father needs to take a more active role with his child and not pass it off to a girl friend.

I would certainly question the violation of privacy rights by any school that used email correspondence. A "girlfriend" who is not the biological parent. Has no legal or ethical right to information of minor children. If the a parent gave permission, and the school released such information, the other biological parent has every right to object. And the legal right to have it stopped.