Possibly, depends on the LEDs. Depending on if your LEDs don't have built in diode bridges (makes them function without polarity), you could simply flip the bulb around. They would not come on with the parking lights, only with the turn signals. Downside to that, is that during the daytime, your parking lights could flash with the turn signals, although at a much dimmer brightness level.
Now the solution I mentioned before, with the switchback modules takes most of the thought out of the equation. There will be 3 inputs, (turn signal, parking lights, and ground) and 2 outputs for the bulb.
Currently the 2 signals use each other as a ground point, instead of having a dedicated one, which is what the switchback module adds for you, separating the signals from each other.
The challenge comes in determining how you want to physically hook the module up. When I did this mod the first time, I bought the Delphi connectors to go between the 194 harness, and the 194 bulb holder.
They are pretty pricey, so when I went to do it the 2nd time around, I went with a cheaper route, and bought some empty 194 bulb bases, fed wires through the bottom of the base, then filled them with silicone to keep the wires from shorting internally. Fed the wires into a 194 twist lock pigtail (I think they were for a Ford), and used an exacto knife to shorten the rubber twist lock teeth to fit into the headlight. I can't find any pics, so not sure if I took any of that part, but it was WAY cheaper than getting the Delphi connectors.