I'm sure Chris does care about her in his own autistic way. That's not to say that he's not selfish about it, but I'm sure there is
something there.
As for care giving, n-n-n-n-NO.
Barb obviously has health issues but she's not at hospice level. Chris can care for her, he'd just do it shittily.
Chris is fully capable of getting her food, driving her, and (if need be) help her get dressed and we all know about holey buttocks. But we all know getting him to do this is a bitch and he wouldn't do it well (i.e., getting McDonalds instead of actually cooking). If you can care for yourself, you can care for a parent to some extent, barring something like dementia, serious illness or injury, etc.
After all, it's pretty sad but you don't have to be caring or invested to be a caregiver, you just need to get the job done.
Echoing
@Gunslinger, if Barb starts to go downhill with dementia then all bets are off and Chris is even more incompetent than usual.