It is somewhat of a problem if the lamp does not illuminate with all the remaining lamps when the key is in the #2 position.

The lamp is fed power whenever the key is in #2 or #3.

It is grounded one of two ways:

1) Through the cluster. From the lamp, through a diode, and to the charging circuit via the charge indicator lamp in the cluster. If there is no alternator output, then the bulb will illuminate via a ground in the alternator.

2) Through the coolant level switch. The switch closes and sends current to ground. It shares a ground with the washer fluid level switch. The coolant switch has a brown wire with a yellow stripe coming from the cluster, with power. The switch has a brown wire which is ground.

You can do some tests at the coolant switch to see if you have a proper ground and whether the switch works correctly.

However, the lack of illumination when the engine is not running is still a question?? Bad diode that won't allow current flow??