There is no way to access deleted messages on the phone. The only slim chance you have is if there is a backup of the device while the messages were on the device and you restore to that backup. That will, however, take the device back to that date/time and anything done since then would be lost.
There is no setting/function on the iPhone for parents to recover deleted messages. Actually, no one could recover deleted messages on the iPhone, except for the method I described above. If there is no backup of the device while the messages were on the phone, they are gone. There is no trash bin/recycle bin like there is on a computer.