Anybody aware of the relationship could call the police... including a school counselor, friend of hers, or other relative. There are no limits on the way police may obtain information.
Any random person could tell officers or report a tip about the sexual activity. When someone faces criminal charges, the victim will have some say as to what the prosecutors do. That being said, prosecutors prosecute cases for the state and not just what the victim's wishes are. They are not obligated to listen to the victim and can actually force the victim to testify.
Her parents cannot "file" criminal charges against you. The ONLY people who can initiate a criminal case against you are prosecutors. Cops can arrest you and send the prosecutors their report. And, anyone on the planet can call the cops and tell them about you shagging the 16 year-old.
Any citizen who knows of your dating may notify the authorities of the possible crime as your relationship may warrant an inverstigation of a de jure statutory rape.
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As you can see from the other contributors, anyone may report you to the police. Even if you break things off with your "girlfriend" you remain criminally liable until the statute of limitations runs. Basically, you are putting yourself in a dangerous situation that could have serious longterm repercussions.
As the answers above explain, if the information finds itself in the hands of the police, there's a possibility you'd be charged with serious crimes. You should also be careful where you post information -- people have been arrested for what they post online if enough information can be gathered.