AFAIK type in index already has partent chain path with ids - but for denormalization you can try to include StartPgeId field or so in indexable fields.

SearchClient.Instance.Conventions.ForInstancesOf<PageData>().IncludeField(x => x.StartPageId());
..

public static string StartPageId(this PageData content)
{
    // travel by ParentLink to find top most parent (StartPage) or fetch StartPage from repository directly
}

Then you can use this field to query for content for specific site.