{musuko no nayami wo kiku} no mo papa toshite no katsudou da 息子の悩みを聞くのもパパとしての活動だ {Listening to [one's] son's worries} is, too, an activity [one would do] as a father.
ryakushite papakatsu da na 略してパパ活だな Abbreviated papa-katsu.
papa toshite no katsudou → papa-katsu.
sore, imi chigee kara nido to tsukau na yo それ 意味ちげぇから二度と使うなよ The meaning of that is different, so don't use [it] a second time. (literally.) That's not what [it] means, so never use [it] again.