As if it isn't obvious, the PO modified the wiring. He set up the ignition switch so there would be power on the Parking Light position.
When the remote fuse block was removed, it should have been a simple thing to remove the four wires, and install the fuses in their proper places. But there it got complicated - the lights wouldn't come on. And with wires landing in non-standard places, it wasn't possible to get it back to that configuration without understanding what was done (and why). We mucked around in there a little and managed to get the lights working again, but now they don't turn off when starting the engine.
Here's what I would do: return the wiring to stock, and then modify from there.
mspa - a data point that may have been missed - that circuit connection board is color coded to accept wires of those colors in those spots. It's properly wired when all of the correct color wires land in their matching color coded spaces. It can't get much easier than that!
The problem is making it nice and neat, with all of the wires swirling around the outer edge of the bucket, leaving the center free for the headlight - and being able to see and check out the connection board.
Wanted: Dead, smashed, crashed or trashed gauges
BMW GAUGE REPAIRS - TACH*SPEEDO*CLOCK*VOLT METER *PODs & LIGHT BOARD*