Some suggestions I have are:
I like the web site design, I think it conveys the simple "child-friendly" look. However, the colors are stark and a bit bold, slightly desaturating/lightening them might be a more pleasing to the eye.
A family member of mine is having a hard time keeping their young daughter from accessing stuff she shouldn't be seeing on the internet. The problem is this little girl is quite a hacker, and has found very clever ways of circumventing the protections, even going to the extent to sneak onto her moms computer to get the passwords to disable the controls. I'm not familiar with the details of your protection schemes, but I guess my advice is dont assume kids, just because they are young, cant crack your stuff! Kids are much more computer-savvy these days so make sure all of your bases are covered.
Good luck on your YC appliation. This idea is refreshingly new and I like it.