On my '01, the rear light is on in both run & park key positions (or is it Run & P ....or...On & P.... old age, not running back out there now).
So, since the lights are not on in the run position, could be multilpe things.
ChecK:
1) like suggested above , battery connections. Look for corrosion too. There is a possibliity that with a bad connection, there isn't enough current flowing to allow all the lights (including headlight) to work, but in park, only one bulb, it has enough.
2) Check the connectors coming off the key switch. Maybe loose connection??
3) In run, jiggle the key and watch for lights. Idea there is switch going bad.
4) Check the connectors under the tank, but not sure on this one since the rear light works in park.
5) re-re-check the battery with bike in run. Maybe you got a dud
6) Next step is to start tracing power thru the wiring harness and find were it stops.