How does the first-person narrative used in "A Father's Story" cause the reader to sympathize with the events and narrator of this story in a case where the events that transpire might otherwise cause the reader to be judgmental toward the narrator?

In general, first-person narration connects the reader the best with a single individual in a story. A reader begins to feel like they are inside the head of the narrator and they are in fact living those actions that they take. It becomes much easier to distance oneself from a character's actions if you're not experiencing their thoughts and emotions while the events transpire, because you can see their rationale behind the actions and feel the emotions that led to those decisions.
When Paul decides to cover up his daughter's accidental manslaughter while driving drunk, you can feel the anxiety he experiences, as well as the love and devotion he has for his daughter. While he makes a terrible decision, it is much harder to judge him when you feel his emotions in that moment.

The first-person narration causes the reader to identify with Paul long before he makes the decision to cover up the murder his daughter committed while she was driving drunk. Because the reader sees his life and events from his perspective, they are already inclined to identify with him and have sympathy for him even when he does a terrible thing.
It's clear that the right thing to do when Paul finds out what Jennifer did is to call the police and medical personnel to see if the man's life can be saved. Instead, he goes out himself to investigate. When he finds the man dead, he covers it up and hides the truth. He does it to save his daughter. Since the reader already identifies with the man and understands a parent's love for a child, it's easy to see why they would feel sympathy with him. They're experiencing the story through his eyes, his thoughts, and that makes them understand his choices.