Stupid question - Have you checked the bulbs?

We've two peugeots and this is the only make of car that we've had
where I found it's quite common for both bulbs to blow at the same
time.

looked at your relay for the lights.

It's the dim-dip resistor or it's associated relay. I don't know where
they are on a 406 though. The dim-dip is designed to stop you using
just side lights when the engine is running. IMO it's implementation
is the biggest pile of cack ever conceived since it is not "fail safe"
as you have found out, when it fails you get two unusable lighting
states for normal driving.