[V]

could also be issues with the DRL module.

first step, check for a proper ground at the socket terminal, not that its trying to ground back through the parking light wire.

Next, if the ground is good, then the issue is in the DRL module or even before that and between the DRL box and the multifunction switch(turn signal lever).

the drl uses the same filament as the turn signal so if its not blinking, it means its power source is not getting interrupted. and logically, if the drl is on, the wire back to the drl box is intact and trouble free.

youd need to then check if the drl module is even receiving the blinker signal input. to check that youd need to test the proper wire in the corrector.

drl module is a small black plastic box, a little thicker than a deck of cards behind the radio.
the needed turn signal wire is in the Connector C1 (8-pin black)
Wire is in Terminal G - (Dark Blue with White - right turn signal input from turn signal switch)
you can unplug it a use a test light or voltmeter to check that with the right directional on, you should get 12v there.

if the drl module is getting the signal but light not blinking, replace DRL module.

if it is NOT getting the signal, then work back to the multifunction switch... and youll have to check those wires too, however i believe youll find the issue before this setp. if not, then you just need to get a wiring diagram of the multifunction switch outputs and test for that directional.