Weighing in on this more out of finding it fun than any serious argument.
I think that both Kirito and Asuna would be generally pretty protective of Yui, with Asuna being more codalling but willing to let Yui experience the emotional knocks of childhood, while Kirito is more willing to let Yui explore and take her physical knocks (the little ones that are part of growing up, you bet your ass Kirito and Asuna be their with swords in hand if Yui was ever in real trouble).
My reasoning for Asuna being the more easy going parent only extends to very specific things. Yui being friends with a boy, one who is mannerly and respectful, probably wouldn't bother Asuna too much, and she'd probably encourage any sort of healthy friendship between Yui and other children who will be her peers as she grows up.
And if Yui starts to gradually grow fond of one of these boys, who is still mannerly and respectful, and growing into a young man, I don't think Asuna would intervene so long as channels were kept open and Yui remained honest with her.
I view this as a consequence of Asuna not wanting to have the sort of distant relationship with Yui that she seemed to have with her own mother. Instead, the two are closer and confide more in one another as a matter of course.
Kirito on the other hand, while usually very laid back and reasonable, seems to have a Berserk button when it comes to Asuna, Suguha, and presumably Yui. It's not that he doesn't like Bardiche, it's that Yui is
his little girl and therefore Bardiche must die (Or accept a punch to the crotch, take your pick
) .
