There is a parking lot on the east side of Newark Penn station, and various private lots in the area (although not as close). If your goal is to visit New York City, and you don't care to actually ride New Jersey Transit, then a better option for you would be to park at the Harrison PATH station and take the PATH in. Less expensive parking and fare; more frequent service.
Secaucus Junction has a large parking lot available, and you don't need to drive through downtown Newark since it's right off the Turnpike. Others suggest parking at Metropark Station (junction of Turnpike and Garden State Parkway) for a longer train trip and cheap parking.
As far as parking in the lot on the east side of Newark Penn station, it's an Edison Park Fast lot. More info here:
http://www.parkfast.com/newark-penn-sta ... th-parking" onclick="window.open(this.href);return false;
For Secaucus Junction:
http://www.parkfast.com/secaucus-parking" onclick="window.open(this.href);return false;
For Harrison PATH:
http://harrisonparkingcenter.com/rates.htm" onclick="window.open(this.href);return false;
For MetroPark:
http://www.njtransit.com/rg/rg_servlet. ... Station=83" onclick="window.open(this.href);return false;