Start circuit on that is not run through any relays other than the solenoid on the starter. Power before the ignition switch is shared, so the fact that it'll run, means that it's not a fuse or anything like that.
It's either the ignition switch, inhibitor switch (on the side of the transmission, like you pictured), or a wiring break. Get out your multimeter and see where the power stops. The inhibitor switch has seperate contacts, so I've seen them fail in "Park", but still work in "Neutral". So if that doesn't work for you, I'd be inclined to ASSume ignition switch, but test it.
Diagrams: http://jdmfsm.info/Auto/Japan/Subaru/Legacy_Outback/1995/1995 Legacy Wiring Diagrams.pdf
Start circuit is on page 15. Super simple circuit.