IIRC, nuCarol does talk about being raised by her dad, and wondering what made the man who taught her everything she knew turn into a psycho admiral...
Dad was a psycho at work. At home he was OK.
In the movie, she had the misfortune of running into him at work.
Well, everyone in the military has two jobs-the one they talk about and the one that they don't.
It sounds like Admiral Marcus was quite open about his active work with Starfleet and his eagerness to see it succeed at protecting the Federation.
I certainly don't find it too far of a stretch to have him be a very doting father and still keep his Section 31 activities quiet.