"The day my dad and his brother stopped me—if you've heard this story, you know how it started—was on what they thought was an uneventful drive."
Here's how the punctuation works:
* Em dash: The em dash sets off the parenthetical clause "if you've heard this story, you know how it started."
* Comma: The comma after "brother" separates the introductory phrase "The day my dad and his brother" from the main clause.
* Period: The period at the end of the sentence marks its completion.
This version is more readable and flows better.