I think specializations should normally be synced as well. It's easier to visualize when using an example like Dog and Cat, which are specializations of "Pet".
If you're syncing "Pet", to me, it would make sense to also sync Dog and Cat entities. After all, both of these are Pets, they just have some extra information. This should be easily tested though by checking what is in your local database directly after a sync.
Edit: starting to question what you wrote since you are talking both about parents and children, but also inheritance. Are you talking about generalizations and specializations or about associations?