Your wife is asking you to solve a problem for her. It's your job to do that if she asks, no matter how rational or irrational it is. Failing to communicate directly to her is the problem, not so much the fact that you think she is wrong.
She wants you to be a leader and is asking you to do something important for her. So talk to her about it. Sit her down and have a rational discussion. If she raises her voice, you end the discussion and tell her you will talk to her again when she calms down. If she fights with you, do the same. Explain to her your point of view. And talk out how potential alternate solutions can be available. But in that, you already need to have the alternate solutions in your head.
For example - is there anything in the alley that you can move? Like garbage cans or a grill - basically is there something that you can move around from your possessions that make it easier for her to navigate the alley. Is there something in your garage that you can move around the same way so that she can enter it at a different and easier angle?
I like the idea above of actually sitting in the car with her and showing her how to get in and out of the spot. Playful banter during is a must.
Finally, talk to your neighbor. Tell him that your wife is concerned about not being able to navigate the alley with his car there and see if there is something he can move (garbage, grill) on his side to make it easier to fit both of the cars so that she isn't so nervous about hitting his car.
Is there any chance she doesn't like her car? Has she been hinting to you that she would like a new one or a different one? This could all be a passive agressive way for you to get her a new car. Maybe something easier for her to drive?
This has the potential to be a good friday thread. But it's Wednesday. So have something good ready to go on Friday as a follow up for us.