I've actually been wondering something as I read through all the comments, and it's unclear from the original question. Yes, the kids have multiple passwords to multiple sites, but does each child have a unique login/password combination for each site? Around here at least, it seems that the schools and libraries have one institutional subscription with a login/password to each (paid/commercial) resource, and that gets divulged to and shared by all students/patrons using that site.
If each child has his or her own set of login credentials, many other commenters have pointed to various "password wallet" type solutions.
If this is instead set up as I put forth above, I suppose the students could still use one of these apps and just put the school's credentials into it. But I think a better solution in this case would be for the school to set up a private portal (VPN, website, or whatever), give each student and anyone else who needs access a single, unique login, and have links through said portal that redirect to the various external sites with the correct credentials. That way you're only asking users (especially younger children) to remember - and keep secure - a single username/password combination.
As the parent suggests, access can be IP based, and by connecting to the portal/VPN, your device at home appears to be coming from the school. That's exactly what the university I work at does, for example if I want to access various online journals from home. They even have it set up so I can initiate the VPN session by visiting a website.