I think Slavy is pretty close here. That B/W wire is sort of a switched ground in the system with all the interlocks. Try to jump the solenoid terminals (the 10 MM nuts) and check to see if you have 12 Volts on the 2 smaller Y/B and B/W wires when you hear that click.
However, I'd almost bet a case of beer that you don't have a bad starter solenoid. Too much coincidence after just draining the battery in the park position. I think something else happened during the initial "what the hell is wrong" testing

Also, you mentioned that you jumped it, it idled, and then died when you tried to drive it. Did it idle normally or did it idle rough?
Do you also have a nuetral light on the speedo when you try to start it?
Do all the lights and horn work?