The whole premise of Snapchat is that there is 'no record' of the pictures and messages sent or received. You do have the option to screenshot the photo but the sender will be notifed. This makes it difficult for parents to monitor what their children are doing...until now.
There is an app that's been out for awhile but is just now gaining popularity called mSpy. It allows the user to see everything their target is doing via their phone. Pictures, messages, emails, you name it...it's available!