First reaction is that the tail lights (rear parkers) are the wrong LEDs. They're not supposed to be too bright, so that they're not mistaken for brake lights. Brake lights, on the other hand, must stand out clearly. The difference is supposed to be considerable.
The very first thing I'd do is get a battery pack (like a jumper pack) and connect the black lead to the chassis. Then, touch the brake pin of the trailer's plug (pin 6, or the second pin from the right as it sits on the car) and make sure that the brake lights come on. Then touch pin 7, which should be the second pin from the left as it sits on the car, and see that the brake lights do NOT come on, and the parker/clearance lights DO. If the trailer's working fine, there might be something wrong with your car's wiring - test lights are handy here, the earth is pin 3 (the middle pin). That should be connected to chassis of the car, and I'm just thinking that if it isn't, power might flow through the parker circuit into the brake circuit and get an earth that way, so both parker and brakes come on - it's worth testing that pin 3 has 0 ohms of resistance to the car chassis.
If that doesn't work, there are two solutions that you could try that I can think of.
Cheaper: put a resistor in each of the lights on the parker wire. If you put it in the trailer plug, you'll also dim any clearance lights. If you have no clearance lights, whack the resistor in the plug and away you go. Not sure of the best value resistor to use, you want something that's going to drop the voltage enough to make the brights dim by at least half, but that's not half the voltage - it's probably more like 9V that's needed. Someone better with resistors should be able to tell you what's needed to do that.
More expensive: replace the tail lights with ones that have a decent differentiation between tail parkers and brakes.
Naturally, fault-finding is always best first, before spending money!