I don't think that's the case, but happy for someone better qualified than me to pitch in. I don't believe parents pay for their children to attend such a school.
I've read articles on schools that opted to become an academy to get themselves away from political interference or to provide more specialist education for their local area. In years gone by there was a feeling that the UK education system wanted all of the children to be academically smart, to become doctors, lawyers or financiers. All to the detriment of practical trades like building, plumbing, mechanics or electricians. There a big shortage of such skills in the UK, whilst there are thousands of young adults with university degrees that cannot get a job in their chosen area due to an over-supply of graduates.
Academies can focus on practical trades rather than academics due to the demographics in their area. Sometimes a school is seen as a failing school as the pupils don't get high grades in traditional academic subjects like maths, sciences or English literature, but actually the children are clever in a different way (practical skills rather than academic). So the school can opt out of the local education authority, becomes an academy and provides better/more appropriate education for the children in their area.
The French, for example, has a similar system that streams pupils down the best route for their abilities, rather than try to push all of their children down a single path.
Wikipedia has some information on Acadamies - https://en.wikipedia.org/wiki/Academy_(English_school)
A bit of a ramble but hope this makes sense.