The Department for Education wants any information on the content of Sats papers removed as pupils take the tests at various times over two weeks.
Officials have been messaging parents since Monday asking them to remove tweets revealing question details.
The DfE said it wanted to clamp down on cheating.
Parents who complain online about the content of papers taken by their children can inadvertently give an unfair advantage to pupils who have not yet taken the tests, a DfE spokeswoman explained.
One parent, Matt Thrower, received a direct message from the DfE after he used Twitter to query whether a part of his daughter's maths paper was relevant to the subject.
The DfE told Mr Thrower: "Hi, can you please remove the tweet referring to Sats? We're trying to maintain the confidentiality & integrity of ongoing tests."
He hit back, saying he was: "Happy to, once you remove such absurd and pointless questions from your tests."
The DfE's official Twitter account warns: "Some children will be taking the KS2 tests next week using timetable variations. Please help us to keep the test content secure. Thank you."
SATs tests are taken by 10 and 11-year-olds at the end of primary school and by six and seven-year-olds at the end of infants' school.
The tests for six and seven-year-olds were the subject of protests by parents last year and in March the government announced plans to scrap them.
Government officials are waging a twitter battle with parents who tweet questions from national tests being taken by primary pupils in England.