I do liberty training and I love it!!!
I don't use Parelli at all, as I am not that fond of it - but let's not go there.
I use clicker training - which I think is what the lady in the second video is using due to the pouch she has around her waste. And, I know Carolyn Resnick has talked about treats during training as well.
You do not have to use treats to do clicker training, although the horses will work harder for food.
I have a section on my website I started about Clicker training. It is not finished, but it goes into equine learning theory and what positive reinforcement is, etc.
I also have some videos on there that demonstrate more about how it works and what you can do with it, etc.
If you are interested, here are some links: Equine Clicker Training Equine Learning Theory
This is the page that lists the videos I have up right now - maybe you will see one that is interesting. Horse Videos
Also, I know that Carolyn Resnick has a video available and is working on some more. I haven't gotten it yet, but it looks good. I have been waiting for her new videos since it says they are out soon.
Have fun with your horse.
By the way, is there something in particular you want to do with liberty training? Maybe I can help.