[Piwoslaw]

Could you explain how and why that works?
How are your DRL's wired to the car's electrical system?
Do you turn your engine off with a kill switch, or by turning the key in the ignition?

Here is how I wired my LED DRLs:
The (-) is just from mass (chassis), the (+) is from the cabin's cigarette lighter socket, which is energized whenever the key is turned, independent of whether the engine is on or off (*). From there it goes through an ON-OFF-ON switch, so that they can be manually forced ON or OFF, or in AUTO mode (the other ON). Auto mode means that the DRLs turn off when the car's headlights turn on (this was done with a relay getting a signal from one of the headlights).

If I turn the engine off with the key, then the headlights (and sometimes other electrical stuff) goes off after 0, 15, or 30 seconds (so-called 'walk home' function, this can be programmed in the computer). But if I use the kill switch, then the headlights and most other electrical accessories stay on for at least 10-15 minutes (they do turn off after a while). But the ECU knows that the engine is off, since a few things do go off, like the ventilation fan.

I think in your case you need to find a wire which does not power down when the engine stops.

(*) My car has a so-called "economy mode" in which the power to all accessories is turned off after 30 minutes of use if the engine is off. This means that if I've used the radio, or the key has been turned in the ignition, for more than 30 minutes after the last engine-off, then my DRLs won't turn on until the engine is on. This is almost never a problem, though.