It doesn’t anymore, but it used to for pretty much all phone carriers in the US.
And even now, on T-Mobile (as that’s the one I use, so the only one I can verify myself), if you have an account with multiple lines (e.g., a family plan), you can go into your account, click “Usage”, then “Text messages”, and it will show you all text info for all lines on the account (but no actual text content). And not just for “kid lines”, but for all regular lines as well. You can look by individual line or download that data as a bulk file.
I just checked my t-mobile account, and despite it not showing the text content (which t-mobile certainly has access to, unlike imessage; t-mobile cannot even track metadata for those individually), it shows an entry for each text with the phone number with info on who was the sender vs receiver, timestamp, and other metadata.
Luckily, T-Mobile only shows that I had 8 incoming messages (all of them were just automated verification code texts) and no outgoing messages this month, because pretty much all my messaging these days is either on discord or imessage.
Even without the actual text content though, that metadata is still some very sensitive info that teenagers almost definitely wouldn’t want their parents to track. Hell, I am not a teenager, have nothing interesting in that data (doubt anyone would care to know about existence of those 8 automated verification messages, and neither would I be embarrassed if someone did), and still absolutely wouldn’t want anyone else to be able to see that info.