Starter won't turn over until I jiggle the gearshift on 1992 Toyota Cressida

Power comes on to accessories with the key on however when the key it turned all the way to start no starter activity untill the automatic gear shift is jiggled around in park.

It sounds to me like an electrical issue between your key and the starter. More than likely the car electrical problem is near the shifter area. I hope this helps.
2 more answers
Some of the possible reason or remedy may be:
a)idle air control valve may have problem, to be repaired.
b)air passage might be blocked with excessive carbon
c)Sometimes proper cleanup and spray can remove the dirt and may be a tricky solution
d) fuel injection system may be clogged, solenoid and catalytic converter may be blocked

All these solutions may help you build enough fuel pressure to resolve start-up problem.

Transmission repair services Vancouver
I have the same problem. When I direct wire the starter to battery it turns over but not with key. Therefore I suspect the park/neutral safety switch its located by the gear shift. The car thinks it's in drive or reverse therefore wont start