Yes! Family Sharing is enabled for Playlisty Pro.
You can verify this for any app by looking at it in the App Store, in the “Supports” section. In actual fact there’s nothing the app itself needs to do to support this – it’s just a switch that we (the developers) set on the app’s App Store definition.
If it’s not working for you (and it can be a bit hit and miss) then it might be worth looking at this Apple support page before calling Apple.