defective relays?

If you have the standard incandescent bulbs on the turn signal and parking light lines in addition to your LED strip lights, you can simplify it and use a single relay, with parking light power going to terminal 30, the LED strip to 87, and the coil inputs connected to the turn signal input and parking light input. Whenever either the parking light or turn signal lights are on, the relay will energize and light up the LEDs, but when both are on at the same time, the relay will deenergize and leave the LED off, which is what you wanted. Just use a non polarized relay without any internal suppression diodes. This uses the low impedance of the incan lamps to act as a virtual ground for the relatively higher impedance of the relay.   If you've removed the incan lamps then this wouldn't work-

Thanks for the suggestion. I do have the normal incandescent bulbs in right now, but I may be changing those soon to LED, so everything flashes at the same rate.