Arduino Resetting in Automotive Application
There is no power spike or loss to the PCB. The car is not actually
  running, I am just turning the key on and off.


And did you verify that with a digital scope? I doubt it. 

I don't think you understand how ugly the electrical system in a car really is. If there is a coil or relay switched then there will be spikes. Do some research on electronics for use in an automobile. 

Automotive relays can have a resistor in parallel with the coil to control the level of the back emf when the relay is turned off. You would think that a reversed diode should be used but a diode causes the coil current to discharge too slow and that causes the contact arcing to be extended. A resistor can be used to speed up the discharge but at the cost of a voltage spike on the electrical system.

An Arduino board is not appropriate. There needs to be a lot more protection.And one wire devices are basically an antenna looking for trouble.

https://www.st.com/content/ccc/resource/technical/document/application_note/1f/d7/fc/6d/2e/27/48/98/CD00181783.pdf/files/CD00181783.pdf/jcr:content/translations/en.CD00181783.pdf

https://electronics.stackexchange.com/a/40425/195251

https://en.wikipedia.org/wiki/Load_dump

https://electronics.stackexchange.com/a/163861/195251