Yes, you can use it with Kaspersky, if you want to. I have tried it.
K9 is not very modern, but it works, and it is supported. I know technically savvy people who use it and rely on it.
An advantage to K9 is that it will automatically apply its settings to all user accounts. But Kaspersky is on a per-user-account basis. Also, with K9 you will automatically have password protection, although that is not very hard to set up in Kaspersky.
An advantage to Kaspersky is that it is more modern, thus it can scan inside secure connections (HTTPS sites). K9 can prevent you from going to the site, but if you do go to a seemingly clean site, it won't see what else you are doing in there.
I have found that each parental control app that I have tried will have both FPs, and things that it should have blocked but didn't. Also, behavior is not always consistent.