Is there an iOS SMS Texting app that allows parent to get a record of all texting messages?
I found this app.
SMS Touch that allows you to setup email address to send copy of SMS content.  SMS are purchased in app.
Net Nanny are also bringing out an iOS app to monitor and restrict iOS devices in young persons hands. details  here

NET NANNY FOR IOS includes Filtering (18 categories), Age-based Profiles, Safe Search, Profanity Masking, and supports Wi-Fi/3G/4G.
NET NANNY SUITE FOR IOS includes all the features of Net Nanny for iOS plus Remote Administration, Reports, Custom Categories, Whitelist/Blacklist. Future releases within the next few months will include Time Controls, Custom Alerts, Reports, and device management tools such as Device Locate and Forced Ring. All new features will be auto-updated for existing users.