LAs have little power on what academies and free schools do. They get the blame when things go wrong but do not have to power or authority to intervene.
It's not just parents cheating - sometimes the whole system has anomalies.
I worked at a CofE secondary school in the NW. When it was oversubscribed it caused all sorts of problems. This was before cared for children, SEN children got to the top of the list. It was based on Church attendance. However the school body was made up of 6 main constituent churches.
Each one applying it's own variation of the church membership requirement. The parish priest signing the actual application.
Some would sign anything, others requiring weekly attendance over 24 months. My church had signing in lists and we took a middle line - even this caused problems. People signing in for a friend, some slipping in and then out after signing the sheet.
Some parents would name a priest on their form - one who was long gone or dead.
We abandoned this and simply said you have to be known to the church. So far this has worked - for us.
The application forms were over complicated which tended to get a self selecting population of pupils in the school. To put it bluntly - less able children - with less able parents found it difficult to make a successful application/fill in the form correctly/favourably.
The school is full, it does well, gets good results and pupils look smart and presentable. I left, for unrelated reasons, and went to another school that was a little more 'real world'
The whole application/selection procedure could bring out the worst in some.
It's hard to make it fair to all and not allow sharp practice.