Due to the high cost of college nowadays, should parents save up a college fund before they have a child?

I'm an only child. My parents had me in their mid-thirties and were married for 4 years before I was born. I'm about to go to college in the fall, and my parents have $80,000 saved up for me. I keep watching those TV specials about college kids who rank up hundreds of thousands of dollars of debt and I'm becoming a little bit worried about their futures and their children's futures (and my children's futures). Since college is becoming more and more expensive, should parents (who didn't have their child by accident) save up a good portion of money for a college fund before the child is born?