If your child is too short for a ride and the ride has a Fast Pass the waiting parent will receive a "Child Swap Fast Pass". The first parent goes on the ride in either the Standby Line or the Fast Pass line (if they have a FP) while the other waits with the small child. Once the first parent finishes, the second parent and up to two other people can enter the Fast Pass line. Since this method uses a Fast Pass the "waiting" parent doesn't have to hang around and wait for the riding parent to finish. Instead, they can ride another attraction, or see a show or get a snack or do some shopping while the first parent rides.
If your child is too short for a ride and the ride does not have a Fast Pass (or the FP's are not being issued) then I believe that the waiting parent needs to hang out at the ride and wait for the riding parent. The waiting parent may then be "back-doored" into the ride or perhaps sent up the empty FP line. I guess it would be up to the Cast Member.
If your child is tall enough, but too chicken then the child must stand in line for the ride. Once you get up to the ride the CM will direct the child and a parent to the chicken exit. The waiting parent will then wait for the riding parent to finish and the swap will take place in the ride.
Make sure to ask the CM at the head of the line about how they want you to do a child swap as you just never know how they are doing things on a specific day.