I am generalizing but father/daughter, as in the OP's case, seems to result in more leeway being given to the child, which can undermine the authority role of the father. Whereas with father/son, it seems that being more stern and authoritative is a natural tendency or, at the very least, the social norm. Same goes for mother/son vs. mother/daughter.
In our household, I am the primary disciplinarian, but I have not sacrificed being able to play and be silly with my kids. Another commenter outlined the need for consistency. I wholeheartedly agree and believe that you can be consistent with your kids and still be silly, play and encourage healthy exploration of new challenges.
i dont think that it's simply reducible to gender because there are many other factors at play (birth order, age difference, personality, birth trauma, age/stage of life of both parent and child, etc.). if you can control for all of those other things, then maybe you could test for gender interactions, but then you'd only have data for that set of controls.