I think the neutral/park starting business is just an interlock to prevent preople from starting their car in drive.

So when you push start a car, the transmission is spinning the engine the way the starter would, just using the car's momentum instead. In a manual this is possible because the clutch makes a direct connection between the engine and tranny. In an automatic, the torque converter makes this connection, but the torque converter is a viscous coupling device and not a direct coupling, so there's not enough of a connection between the tranny and engine to rotate the crank.