No. Slow Time Club is for managing your own iPhone. It isn't a parental control app, and you can't use it to set or lock limits on someone else's phone.
Why
When you allow Screen Time access, Apple lets Slow Time Club manage the phone it's installed on, for the person using it. There's no way to link two phones, see another person's usage, or control their blocks from your phone. Whoever uses Slow Time Club can change their own blocks, within the rules they chose.
If your teenager wants to use it
They can install it on their own iPhone to manage their own time. If their phone is set up as a child account in Family Sharing, Apple may limit which apps can get Screen Time access, so they might need your help to allow it.
For parental controls
Apple's own Screen Time, set up through Family Sharing, is built for this. It lets a parent manage limits on a child's iPhone.