What do you think about buying a franchise right out of college?
One of the joys of being the boss is that you have the control and responsibility for making things happen. You are indeed the CEO - but you're also the janitor or dishwasher or any other job that needs to be done at any given point in time. Being successful involves being willing to do any job in the business and often leading by example.
It's possible that your parents want you to work for a couple of years before providing the money for you to get a franchise in order to allow you a chance to gain some real world experience working for a living. It can be experience that will help you manage employees better when you become the boss and can also teach you lessons that will help you avoid expensive mistakes once you become the boss.