Okay I have put leds in my 2002 f150 and after doing so when I turn on my left turn signals the right park lights blink with the left side and the same with the other side. If I turn on the right turn signals the left park lights blink with the turn signals. I have installed a led turn relay so I don't get hyper flashing. I have found that if I put just one regular bulb in any of the sockets this feed back through the park lights no longer happens. I am pretty sure it is the oem trailer light controller that is causing the feed back but I cannot find info for the oem 4 pin trailer light controller. I cannot even find the wiring info for the trailer lights even though I have a haynes manual. There is nothing anywhere about the factory 4 pin wiring. I have a set of canbus decoders on the way and I am hoping they fix the issue. If I put a resistor on the park lights to ground the feed back stops but I do not want to leave a resistor hooked up to the park lights as the resistor will get very hot and possibly melt something. I think the controller is under the hood on the drivers side because I can hear it working when I have the leds in. It is also causing one of the relays in the fuse panel under the hood to click with the flasher relay under the dash. Again the relay under the hood stops clicking if I put just one regular bulb in the system. And yes my cruse control still works without having any load resistor relays.

 

I had this very problem with my 03, I had bought cheap "amazon" LED's and installed them front and rear. I also installed the appropriate signal flasher.
If I had my lights turned on all seemed normal, but with the lights off all the parking lights flashed with the turn signal engaged.
It turned out that the red bulb in the drivers rear had shorted out (in less than a week). I diagnosed this by pulling the tail lights out and replacing one at a time with a new incandescent. I started at the driver rear and as soon as I put the incandescent in the problem disappeared, same as you.

You DO NOT have a problem with your wiring, trailer or otherwise, it is the BULB itself!

If you didn't buy a name brand such as Sylvainia it's a poor quality bulb for sure. I can't say that a name brand will work any better because I haven't purchased any to offer feedback. But if you have to have LED tail lights, I would go with the Sylvainia's or similar.
I returned the rear red LED's and have had no problems since, I think because our trucks use the one bulb for three functions that cheap LED's can't handle the constant use (park lights, braking, turn indicating)
I still have the amber bulbs in the front and they are working fine still (over a year now) I really like those because they are much brighter and more amber than the incandescent bulbs were.
Hope this helps you

 

Thanks for the info. It was the bulbs. I have several batches of bulbs I have ordered from the same online retailer in all my other vehicles with no problems so I took the bulbs out of one of my other vehicles and no more feedback so I just got a bad batch this time. I went and got new ones from autozone and everything works.

 

That's cool, What brand are you using? I haven't put LED's in my tail lights since the issue, to be honest the ones I had purchased didn't have enough difference in brightness between the parking light and stop/turn for my taste so I just left the stock style in place.

 

They are whatever brand Walmart sells. They work really well. They light up a third of the leds for the park lights and then light up the rest for the turn signals and brake lights so it ends up being the same as regular bulbs. My other vehicles I have much more expensive bulbs by JDM that there is a big difference in brightness but for this truck I was doing all the bulbs at once so I didn't want to buy the expensive bulbs.

 

In my F-150, I changed my flasher relay to an EP-27 LED relay. All (cheap) LED lights work just fine.