If the OP’s specific request was asked of me, I would feel a bit uncomfortable for two reasons:
1.) If she couldn’t get the car parked, why would I have more success? And then I’d be responsible for damaging two vehicles that were not mine.
2.) I view a car as a somewhat personal space, so I would feel uncomfortable getting into it from a tznius standpoint. I would have no problem if the help involved something like changing a flat tire, since that would not involve me actually getting into her car.
Also, knowing my personality, if I were asked that question and put on the spot, I’d probably hang my head and stammer something similar to what the rebbi in this story said, and then regret it later. Having the benefit of sitting in a warm office with a minute to think about the best response, I’d probably make a self deprecating joke about not letting me do it for the safety of her car and the ones around it, apologize, and move on. Or possibly offer to help guide the parking from the outside through hand signals, if that were helpful.