Here's the wiring diagram:

So the way they work is like this: When the park lights are turned on, voltage is supplied from the Tail Fuse to one side of the front side marker bulbs on the brown wire. The path to ground for the side marker bulb is through the lt blue wire to the large front turn bulbs on the turn filament. The smaller side markers will light up but not the larger front turn signal filaments. So we know that part of the circuit is OK based on what you observe.
When the turn signal is on, but not the park lights, voltage is supplied from Turn B/U Fuse through the light blue wires to the marker lights (opposite to the case above). They are supposed to turn on since they are grounded through the brown wires to the park light filament in the larger front bulbs. As before, the small park lights will light but not the larger park light bulbs. The turn signal filaments on larger front bulbs will of course light, just not the park filament. So if the side markers don't light under this condition, then either the park light filament in your front turn light bulbs is burnt, breaking the ground path, or you have the wrong bulb in there, or it's improperly making contact to the socket.
If both the park lights and the turn signals are on at the same time, the side marker bulb for the selected side will turn off when the front bulb flashes on, and it will turn on when the front bulb flashes off. This happens because when the flasher unit applies voltage to the blue wire, there is already voltage on the brown wire from the park light circuit, so the bulb will go out since there is the same voltage on either side of the bulb. When the flasher momentarily turns off between flashes, it removes the voltage from the blue wire thus turning the side marker back on again. Once again, if it doesn't work in this manner, the same diagnosis as above applies.
Post what you find.