Akahi is a goody two-shoes so he told his parents he wanted to be a vigilante, out of respect, mostly. He also ignored their protests about it because he thinks he's doing the right thing and they raised a goody two-shoes, not a quitter. It's been a rocky contest of stubborness since.

Shirahime on the other hand, lied to them, because that is what normal teenagers do when they are considering vigilantism. That came to bite her in the ass.