It occurs to me that you may have a dodgy connection between the Chrysler starter solenoid and the safety cut out switch in the gearbox. This safety switch applies an earth to the solenoid only when the box is in either park or neutral ensuring that the solenoid and hence the starter will only operate under those conditions.
(Or perhaps the safety cut out switch in the gearbox itself is faulty, but in my limited experience they just fail, at the moment you’re setting off to drive from Wales to Morocco!)
This connection, solenoid to starter motor, is made by a thinnish white wire with a brown tracer that disappears down from solenoid towards the gearbox.
Alternatively, there may be a dodgy connection between the ignition switch start position and the starter solenoid. This is a thinnish white wire with a red tracer.
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