I've checked with a full household where all five YAs and As (it's an Apocalypse Challenge and I haven't lifted the restriction on moving them out yet!) are either the parents or the infants have been made care dependents. Parenting skill for all of them rises only when the infant is being bathed or washed. Nothing else brings up the skill bar or changes the skill percentage. If it's happening for one task it looks as if it's an oversight that it doesn't rise for anything else – I wouldn't have thought bathing is more important than the other infant skills.
I've also had an eye on it today again:
Feeding with bottle brought a little progress for the mother's parenting skill
All other interactions like talking, caressing, change diapers, bring her to bed and so on had no impact on the skill.
I'm still not sure if it might be related to one of the mods I use as in my last vanilla game that I started about a month ago there was, as far as I remember, a constant progress for any interaction from newborn to kid and because I set the lifecycle to long it built up annoyingly fast.
