Bury Lawn in Milton Keynes is one of 13 fee-paying schools in the UK now run by the Dubai-based firm Global Education Management Systems (Gems).
The international company, set up by millionaire Sunny Varkey, aims to create or take over at least 200 schools in the UK but will charge relatively low fees.
Last year it bought Bury Lawn, where fees range from pound;2,000 to pound;6,500 a year.
But parents have complained about Gems' plans to increase class sizes from 18 to 24 and have set up a website to debate other changes. One said that at least 20 families had removed their children and that others were worried the company was "trampling" on the school's ethos.
"Our concerns are for our own children as well as the children at the other schools Gems has just purchased," he said. "They do not have the experience, infrastructure or cultural understanding."
However, Gems said it had received complaints from only three parents and that most were pleased with its work.
The company has informed families it will wait at least a year before raising class sizes and will only do so if there is enough room.
John Collins, chief operating officer for Gems UK, said bigger classes gave pupils more opportunities. "It would tarnish our reputation if classrooms were over-crowded," he said. Gems's international network of schools employ more than 4,000 staff and teach more than 40,000 pupils.
The firm's UK advisory board includes some of Britain's top educationists and is chaired by Mike Tomlinson, the former chief inspector who is leading the inquiry into 14 to 19 education.
Mr Collins said Gems's schools would provide different levels of service - like an airline with a business and "no frills" economy class. "They both arrive at the same destination," he said.
"Our schools in the higher price range will have better facilities with extras such as swimming pools and better sports facilities than others, but all will have the same standard of teaching."
Parents have also criticised Gems, on the website, for appointing business development manager Fergal Roche as Bury Lawn's new principal. Mr Roche left a previous job as head of St Andrew's prep school in Eastbourne in 2002 after a row over a church service in which a child was asked to lick chocolate from a nappy.
Mr Collins said the incident was "an error of judgement that in no way detracts from Mr Roche's qualities as a headteacher". He added that Mr Roche's decision to step down to take the headship had been a surprise but was excellent news for the school.