It appears the gauges illumination comes on with the parking lights. It's not something I notice since I don't use the PARK position for anything. And I think it's silly to run gauge illumination during daytime running, but hey, that's where government regulation takes us.
Your headlight relay isn't stock, though it may have the same configuration - only cross checking will tell.
The headlight and parking light (in the headlight) are wired separately.
The funny thing is that it's possible to have the ignition come on in the PARK position with just a change of one or two wires - I believe to the ignition switch. Some PO went to a bit of trouble to come up with this solution.
Wanted: Dead, smashed, crashed or trashed gauges
BMW GAUGE REPAIRS - TACH*SPEEDO*CLOCK*VOLT METER *PODs & LIGHT BOARD*