I'm presuming a number of things here:
1) The DRL 'bulb' is used both as the DRL (on all the time), and the indicator (flashes OFF and ON, then goes back to permanently ON).
2) It's essentially just an LED 'bulb' and has a single 12V feed to it, plus it's ground of course.
So, cut the 12V wire and use an FET as a switch in the wire. Use a PIC running a timer fed from the 12V input side - when the 12V feed appears (i.e. you've started the engine) the 12V feed will go high, at that point the PIC can turn the FET ON (DRL ON) and start timing - once the timing goes past a certain point (a bit more than the ON time of the indicators) the PIC should turn the FET OFF (turning the DRL OFF). The DRL will then be OFF the rest of the drive.
When you use the indicators the 12V will vanish (for the OFF time), and then go high again (for the ON time), this will reset the PIC which will start timing again, but won't time out as the 12V will disappear before it has time to. At the end of the indicator sequence the 12V will stay high, and the PIC will time out, and turn the DRL OFF again.
This will give one spurious flash of the front indicators, slightly longer than the normal indicator flash - but that's probably acceptable?.
You could also do it in hardware, it's simply a monostable multi-vibrator, a 555 could be used.