I don't think you can do this with just diodes, as the parking light circuit is just "one" not left/ right
As I understand this, you want the halo's on with parking lights, but blink with turn signals? Here is one way
Separate the halo to left and right circuits, and get two relays
Wire the contacts from fused power and the output of the conacts of each relay to each halo, so they are powered one halo from each relay
Make CERTAIN you check the relay coils that there IS NO DIODE internally
Tie one end of both relay coils together. Wire that one point to the park lamp circuit. If you want the halo lamps on with headlamps wire it to the TAIL lamp connection instead
Now take the remaining end of the coil wires and wire the left to the left turn, and the right to the right turn
HERE IS HOW this works
With the park lamps off, and the turn signals actuated, the relays trigger as if the park lamps are a ground. In other words the park lamps themselves complete the ground circuit of the relays as the signal circuit activates the other end of the coil...........the halo flashes
With the park lamps on, the halos light because NOW the relay coil circuit gets it's ground return through the ununsed flasher filament, the opposite of above
When you use the turn signal the halo will once again flash, but will be ON when the flasher bulb is OFF. This is the way some side markers work that flash. When the park circuit is active, and the turn circuit is active, the relay coil has 12V at both ends and does not activate. So with the park lamp on, and the flasher bulb on, the halo is dark. When the flasher bulb turns off but the park is still on , the relay coil sees ground through the flasher filament(s)