What is my liability for an accident that happened on private property?

Yes.  Your liability for driving your car negligently doesn't depend on being on the public highway.  You can even be held liable for negligently backing your own car into someone else's, in your own driveway.

Parking lots, as private property, aren't subject to all of the rules of the road.  For example, a stop sign in a parking lot isn't the same as one on the street, in legal effect.  However, if you run that parking lot stop sign and cause an accident as a result, it's still negligent, and you'll still have to pay the damages (or your insurance will), even if you can't be given a ticket for the stop sign.

Where the accident occured doesn't matter, UNLESS there was something that the property owner did that contributed to the accident. In that case, one or both of the parties involved in the accident might have a claim against the property owner, as well as potentially against each other.

For example, consider a "dog bites man" case. If you are walking up my front walk--so you're on my private property--and someone else's dog runs up and bites you, I have no liability to you. It's not my fault that the dog bit you, and I had no duty to control the dog. It would be different if it was my dog, of course.

In the case you describe, if you were negligent while driving, you can be held liable and be expected to pay. If there was something wrong with parking lot that caused the accident, the injured party may be able to sue bank as well; and you might be able to sue the bank for their share of the liability, to offset your own. But only if the bank was somehow at fault, vis-a-vis the parking log.