Hmm, I think it's still in parking orbit around earth, no?

Yes, I believe you are correct, I got slightly too excited. (: It's on a highly elliptical earth orbit at the moment. The second big test is doing a long stage 2 burn after passing through the Van Allen belt. I believe that will take place in 6 hours or so.

It's not that elliptical (at least not for another 5 hours or so when burns resume)

It is quite elliptical. Right now the second stage is in GTO, with an apogee of 7000km and perigee (presumably) of ~200 or so. When it arrives at apogee, it'll reignite and go hyperbolic, onto a trajectory that escapes the Earth's SOI - then it will no longer be in Earth orbit.

Driving around looking for a free parking spot

My understanding from the diagrams they showed is that the orbit is a wide one (oval) around Earth that passes near Mars.

It's actually orbiting around the Sun, it passes near Earth orbit at the perihelion and near Mars orbit around aphelion, but is not actually in orbit around either planet.

It has just burned again to raise apogee, and will burn a third time to switch from earth orbit to a transfer orbit (edit: or perhaps the next orbit is the final Earth-Mars orbit?)