I had the same kind of problem: Whenever I insert the contact form code in a post, page or text widget it would not parse the code. But it turned out to be an incompatible plugin, in my case it was ‘HeadSpace 2’.
If you’re having the same problem I would recommend disabling all plugins except Contact Form 7 and see if that fixes the problem. If it does, re-enable plugins one by one to see which one is causing the problem.
If none of the other plugins are causing the problem, switch to the default theme to see if that’s the problem.