Parelli (as well as other bridle-type horse trainers) believes that one should stay out of a horse's mouth until the horse positively responds to the hackamore (or sidepull) and until the rider has an independent seat and does not use the reins for balance.
To achieve more refined communication after success in the hackamore, a snaffle is employed for soft feel and vertical flexion.
Bits are communication tools and should not be used as instruments of force. Any bit can be torture in the wrong hands or if applied with the wrong attitude. Attitude and intention rank high in the use of the snaffle, either positively or negatively. How the snaffle is used can make or break a horse / human relationship. Turn things around: would you want YOUR employer on your back and in your mouth all the time? or flowing gently "with" you and only using the bit for refinement when necessary?
Control of the horse is something that comes from respect, good training, two-way communication; not bits.
A theory attributed to Ray Hunt is that: "If you can't stop a horse with a curb, go back to a snaffle. If you can't stop a horse in a snaffle, go back to a halter. If you can't stop a horse in a halter, ride bridleless."
These trainers are saying that one needs to earn the right to use a bit!
If our Icelandic Horse is naturally gaited, if he's smart, if he's trained well, there will be no need for him to carry the rider's weight on his mouth. He'll be soft, light, flexible, and responsive.