There's no FastPass or slip of paper -- the second parent rides as soon as the first parent (or other adult) gets off. Basically, you all three (or however many) wait in line together, either in the regular line or the FP line if you have FPs. Then when you get up to the CM, you tell them you want to do the baby/child swap. They will tell the parent that's waiting with the baby/child where to stand -- this differs from ride to ride. Then when the first parent gets off the ride, they come back and get the baby/child, and the other parent gets on the ride. As an "added bonus," if you have an older child with you, that child can ride twice in a row -- once with each adult.
Does that make sense?