"In the park" sounded better to me too but it wasn't that big of a change to me to use "at".
'Dad and I' serves as the subject phrase already, so we don't need another subject 'we' in there.
If you need to repeat the subject in English, you would do so in Spanish as well, including the comma: Papá y yo, nosotros escuchamos música."
¨dad and i listen to the music in the park¨ ruled not correct. i find that hard to believe without context.