I agree with your point very strongly.
But I am not sure the study does show its just parents.
I know that at my school, PP children don't do as well as the PP national average and there are schools whose PP children do way better than ours, as this data is available. So the school you go to does have an impact on pp children's chances.
It's to do with getting a critical mass of funding within school to provide meaningful support, if you have low pp numbers its hard to achieve that. So it's partly school funding. The amount per chikd is low so it's only useful added together
But it is is also linked to your point, in that outside agencies centre support in areas with more need overall. So we don't have some of the succesful out of school social schemes that are running in other areas to back us up.