Excellent thread on socialization rethinking "popular" early socialization
Really makes you stop and rethink the socialization that we have all learned to do.
As for what happened today at the park, instead of going to Petco after the park, I would have taken the pup to the outside of the dog park, engaged her in play to the point she was completely focused on you and then ended it on the high note.
By the time she got to Petco and then later at home with your dad, she was already completely overwhelmed. Her stress levels were already high, it isn't surprising the sudden appearance of your dad caused the behavior you saw.
Again, the link posted is a good eye opener.