Still have a job at this point
So next time...
...if you come across anything like this again, let your manager know RIGHT away that there might be issues. As a manager, I hate surprises, but if I have the details on one side of the story before the others present theirs, I can often defuse the situation because I sound like I know what's going on, and it makes me trust my staff even more if they share 'uh oh' moments...
If your manager hasn't brought it up, bring it up. Explain everything, in detail. If he/she fires you on the stop, F*&# em. Good leaders love proactive and transparent employees. If they're a bad leader/manager, they'll throw a fit - which means you probably don't want to work there anyways. Consider it a good test of their leadership skills and how the bank values their employees.
As always, learn from your mistakes. I give my staff members a lot digression when crap like this happens. For them to be proactive helps diffuse the situation quickly. Imagine what a different conversation would of happened if YOUR manager called the daughter about her mother poking about her account and that was communicated to both parties?
Mom: how much does my kid owe?
You: I can tell you, but we will have to let her know that we told you the amount and that you inquired about her balance.
Mom: never mind.
And if you help your manager keep out of trouble, they'll be more willing to trust you with more responsibility. At least a GOOD manager will. It's a two-way street - you find out what they're like and vice-versa.