As far as Harris Neck is concerned it does not come under the National Parks but rather as a 'Wildlife Refuge' under the operations of the US Fish & Wildlife Service which is a separate bureau/office under the US Department of the Interior ; http://www.doi.gov/index.cfm
It should be noted that once again the future of Harris Neck is being decided in the courts...
A friend of mine is with the Parks Service in New Mexico. She has spent the past 25yrs of her career documenting and protecting a site from 'looters' and overgrazing. Yes, she'd love to get her paycheck but right now she's far more concerned about 'the site'. She went into work prepared to continue doing her job only to be told she would not be able to be armed to protect herself plus she no longer had arrest powers under the government shutdown. Rather than risk her life she has gone to visit her parents in hopes that the visit will be cut short. (I hope she doesn't hold her breath)
Yes people have 'jumped the fence' at many of the National Parks;
... and there is bound to be plenty more as this continues. We all have to look at the BIGGER picture though. 65% disapproval rating of both Houses and the latest NBC poll has 95% of people polled saying FIRE the entire lot on Capital Hill!