How do I catch palkia without using my master ball?

Firstly, you'll want to get Palkia down to the least amount of health possible - if any of your Pokemon know the move "False Swipe", you can use that without fear of killing it.

Secondly, inflicting it with sleep or paralysis greatly increases your catch rate, so using moves like Stun Spore or Hypnosis will make your life easier.

Finally, use a dusk ball during the evening hours (when the ground turns visibly dark, after 6PM I believe) for a 400% increase in your catch rate. Doing all of this, I ran through about six dusk balls before finally catching Palkia - good luck~!