I checked the FAQ before I bought a kindle fire for my son. It clearly states that:
You can enable Parental Controls to restrict access to the LOVEFiLM Instant app on your Kindle Fire. When Parental Controls are enabled, you will be prompted for a unique 4-digit PIN password whenever a user requests to start watching a title.
To enable Parental Controls:
Go to the Kindle Fire Home screen.
Swipe down the Quick Settings menu and select More....
Select Parental Controls.
Follow the onscreen instructions.
Stupidly I assumed it was the same pin code as used for lovefilm on the web but it turns out it is the kindle pass phrase (not pin) which means you can only turn Lovefilm off or on!!!
Why when you on the web can set restriction based on certificates??? More disappointed with the kindel the more I use it...