Okay, so I noticed the big difference is my 8 years of service was in the Army National Guard, so I'm not qualified for VA benefits. I get the VA loan and stuff, but not any actual VA benefits. The VA ID card is something that is newly available to veterans like me, so maybe that is part of it.
As for the National Park, they were willing to sell me the annual pass using my DD214 and NGB 22, but they weren't able to do anything about getting an actual Veteran ID card, which is only available through the VA for veterans like me...