The story mocks the idea that everyone can be equal.
While everyone may have equal rights -- and equal rights are certainly necessary -- everyone will not be equally smart, good looking, talented, etc.
Vonnegut mocks the idea that it is possible to achieve this kind of equality. The Handicapper General takes this idea to the extreme by bestowing life-altering handicaps to everyone in an effort to make everyone "equal."
As Harrison Bergeron himself proves, this type of equality is false and doomed to fail. Even his handicaps cannot prevent him from being greater than others.
Bergerons parents satirize those who go along with ideas even though these ideas may hurt them -- when his wife suggests he remove some of the weights around his neck, he refuses.
