Non Canbus led bulbs may be the cause however I recently had a problem on the Z4 with the indicator bulb staying on with side and headlights, turned out to be some melted wires inside the headlamp causing a live short to the indicator bulb hence it staying on. There is also a bulb warning feature that illuminates the indicator if it detects a sidelight bulb out so illuminates the indicator as a backup.
I would put normal bulbs in the sidelights and see if that sorts it, if not explore for a short.
My fault manifested itself some time after I fitted supposed canbus compliant led bulbs, wonder if yours is the same cause?