Power to the front left turn signal bulb is provided by fuse F210 (5 Amp) in the Central Junction Box (CJB). The right turn signal bulb gets power from the CJB via fuse F208. And like joegr pointed out, the bulbs should have power at all times. Ground to activate the parking light or turn signal is from the Front Electronics Module (FEM).
One thing you might try is to run a dedicated ground via a jumper wire and back probe to the ground of the bulb (Black/White tracer wire at the bulb socket or Black/Blue tracer, it's unclear from the wiring diagram which is parking and which is signal). If the turn signal filament lights up this way, you know the issue is either the ground wire between the bulb and the FEM or within the FEM itself. If the bulb still doesn't light, then the issue is with either the power wire to the socket, fuse F210, the contacts for the fuse within the CJB, or power to the fuse. Since you read voltage with your meter/test light, it would likely be a corrosion/connection issue in the supply circuit rather than a blown fuse.
When you measured "0V with the bulb plugged in the socket" what were you using as your ground reference? If using a chassis or battery ground, then I'd definitely suspect a corrosion or connection issue with the supply circuit. The bulb is providing a relatively low impedance path to ground through the FEM and sinking what little current is available to the bulb to ground, causing you to measure little to no voltage at the socket. Since neither the parking light nor the turn signal work, it seems less likely to be the FEM and more likely to be the common supply to both.
The CJB is located in the right hand front footwell and the FEM is located at the left hand A pillar.
-Rod