Earlier quoted context omitted.
Do you know the details of what happened here? Unless you know otherwise, it sounds like your father probably ended up clicking the "renew subscription now" button when prompted, not that simply logging in triggered the charge. I'm ready to believe that the renewal prompt was heavily geared towards making it easy to renew, and Netflix should be required to make it equally easy to cancel, but unless you know he didn't…
No the onus is not on the customer to not step on the landmine.
I've logged into a cancelled Netflix account before, and while they definitely want you to renew, there's a very clearly marked button that includes the renewal price. If OP's dad absent-mindedly clicked that button, that's a very different story than if simply logging in and navigating to a movie issued a new charge.