Offer (and advertise) a package of Metro fare/parking for getting to and from the National Zoo on weekends from any metro parking lot in the system for a family of a given size. Price the deal such that the entire family (say, of up to 7) gets to park one car and ride the metro both ways for a price that would be cheaper than paying for zoo parking and the cost of driving in. Advertise it as a great deal (make sure its actually a good deal) and a fun experience for the kids, and faster and less stressful than driving.
There are enormous traffic backups and parking congestion on weekends around the zoo, and capacity on the weekend metro is there to add riders (at least once SafeTrack ends). Why not leverage that capacity and encourage families to take advantage of Metro to have a safe, fun, and less stressful ride to the zoo? This doesn't actually cost anything beyond the advertising, which can be done fairly cheaply one assumes-- it would be a pricing program designed to increase ridership. Someone has to do the math of course to figure out what to price it at and what would make sense, but giving families an incentive to use the metro to get to the zoo is a net plus for everyone.