TK isn't going to be an option. They don't fly to Australia, and you're extremely unlikely to find a reasonable fare going in that direction. PWM to SYD is 10,130 miles; via JFK, IST, and SIN -- the most logical routing I can think of in that direction -- it's 14,584 miles. Airlines aren't all that keen to fly somebody an extra 4,500 miles without charging -- substantially -- for it.

If you need the PQD, I suspect your true options are UA and NZ. Anything else, you're going to have to piece together using multi-city search, and you're going to have to deal with plating restrictions -- between those two factors, I suspect you're going to find that prohibitively expensive.

If you don't need the PQD, I suggest you play around with Google Flights or something similar to see what your realistic options are. EVA doesn't fly to SYD, and they don't tend to play nicely with others in terms of ticketing agreements. I think you'll find that's unlikely to be an option either. On a few sample days that I checked, Asiana seemed to be the most reasonable option in terms of cost.