I think you misunderstood parent's (admittedly poorly illustrated) point.
As someone who grew up mostly before cell phones, it forced a greater level of planning, responsibility, and freedom on me than kids now normally experience.
I'd often call my parents (gasp! Remembering my house phone number!) to adjust plans, by telling them where I'd be for how long. And generally, they had no problem with it.
I laugh thinking about the absolute fucking nuclear meltdown a lot of helicopter parents would have today at middle schoolers saying "I'll be over at this friend's house for the day. Will give you a call closer to dinner. If you want me, call their house phone, but we might be out in the neighborhood or woods."