If a parent currently refuses to allow their child to be patted down by a stranger in uniform that happens to have a job at an airport checkpoint, they may be arrested or fined, or - at the very least - will miss their flight. It doesn't appear that this changes with Rep Chaffetz's new bill. However, this bill would address a concern that he has based on what happened to his own daughter: she was patted down without his wife (who was traveling with their daughter) knowing where she was. (Here's the full text of the proposed legislation.)
Moral of the story: if you have children and don't want to have them confused or worse by an invasive custody search: Don't fly!