There are two different "programs", so to speak. The first is the Rider Swap. This is where the entire party waits in the line and then one rides, while waiting for the other watching the little ones (or the big ones who get scared to go on). When the ride is over, the parents switch and up to two other guests can reride with the other parent.
The Rider Switch is the paper ticket that allows one to come back at a later time to ride after the one parent has ridden. These are only given if the child does not meet the height requirements for the ride and must be measured by the attraction CM (sometimes having to wake a sleeping child).
You cannot do a Rider Switch if the only problem is that the child is too afraid to ride the ride, but meets all the other physical requirements of the ride. You will all have to wait in the SB line or the FP line if all have them, yes, that means you would have to burn an FP for the one who is afraid to ride and do a Rider Swap.