I just expirienced the same, but can't confirm it from an old savegame I played before. I'll try to test some more tomorrow, but as it's been different to my last household, there are two options where it might come from:
1. My actual household is the first one I'm using mods - I'm using quite a lot of CC furniture and also some script mods (MCCC & Woohoo, Slice of Life, Meaningful Stories), so it might also be caused by one of the mods or a setting in MCCC
2. There are further sims in my household, I'm not sure if it's only not working for other members of the household who take care of the infant but also for the parents
As I don't own the growing together extension it must be related to the parenthood DLC.
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.
