Did you see my post about the Headlight relay in your original thread? (Yesterday's thread is here.) It can cause just what you discribed. I said there, that the Fog lights were a separate issue. (Also the right side FL is usually not equipped on American 240s, so don't blame the PO.)

But you may have other issues too, because the Fog Lights are not tested by the Bulb Failure Sensor. This may be the subject for another thread, once the original problem is fixed.

Please check those rear lights with the light switch in the Parking Light position (leave ignition key off). My hunch is that they will be On.

Then start the car and check them again, this time with the Headlights switched On. If both sides Front and Rear Parking/running Lights are now off, it's the relay.
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Bruce Young
'93 940-NA (current) — 240s (one V8) — 140s — 122s — since '63.