Public places wouldn't stop my parents from putting me in check (not that I need it
). Even if they toned it down, "the look" would be enough for me to want to disappear.
Keeping your cool (as the child) is number one. If you act like an immature teenybopper, they'll discipline you like one and won't take you seriously. Having everything in your head planned out helps for when they have a rebuttal. Parents are pretty predictable, so when you're organized and know what to say, you won't sound as stupid.
Also, of course, making loads of du'a always helps; for Allah to make your task easier, your parents more understanding, and for the best outcome. Musa's (as) du'a before approaching Fir'aun comes to mind.
I remember my sister told my parents that she was about to drop a few classes (that she was failing), ready to change her major (again), and just totalled her second car, at the hospital right after she got into a car accident. They couldn't freak out after she was "traumatized" at the hospital. A part of me still thinks that she had it all planned out...