LED's will simply not work in side marker lights without modification. If you will note, there are only two wires going to the side marker lights while there are three going to the front marker (parking) lights. The missing third wire is the ground wire. Two decades ago it became popular to use a "floating ground" on side marker lights on a lot of vehicles. BTW the JK markers are just that, they don't also act as flashers with the turn signals).
When the parking lights are off, the side markers (with incandescent bulbs) will flash in sequence with the turn signals. That is when the front turn signal is lit, the side marker light is lit. When the front turn signal is off, the side marker is off. With the parking lights on, the side markers (again with incandescent bulbs) will flash opposite of the front turn signals. That is when the front flasher (not parking light) is on, the side marker is off, and when the front flasher is off, the side marker is on.
This is because there are two +12v wires to the marker and the fact than an incandescent bulb will pass current in either direction. When the parking lights of off the +12v for the flasher goes in on the flasher +12v wire and goes out the parking light +12v wire to ground. When the parking lights are on, the +12v for the parking light goes in on the parking +12v wire and out the +12v to find a ground. However, when the parking lights are on and the flasher is also on, the side marker goes out. This is because when you have +12v on both wires, they cancel and no current flows. Thus, the side marker alternates with the front flashers.
However, an LED (Light Emitting DIode) being a diode can only pass current in one direction. Thus when you have an LED bulb in the side marker you will have either a flasher (when the parking lights are off) or a side marker (when the flashers are off) depending on which way the bulb has been installed. To fix that you have to insert f full wave bridge rectifier in the circuit before the side marker lights. The easiest fix is to remove the LEDs from the side marker lights and go back to the incandescent bulbs.