Gavin Robinson claimed it was the “same old story of half-truths” as he demanded answers on recent controversies.
It came as Northern Ireland Secretary Hilary Benn said the saga risks damaging trust in politics.
Sinn Féin has been criticised over its handling of scandals linked to former press officer Michael McMonagle and ex-senator Niall Ó Donnghaile.
McMonagle recently admitted to a series of offences, including attempting to incite a child to engage in sexual activity.
It subsequently emerged that two other press officers, who have since resigned, provided job references for McMonagle after Sinn Féin had suspended him.
The references enabled McMonagle to get a job with the British Heart Foundation three months later, with the charity unaware of the allegations he was facing.
Then this week, Ó Donnghaile, a former Belfast Lord Mayor, revealed he had left the party after sending allegedly inappropriate text messages to a teenager from Northern Ireland.
Subsequent media enquiries revealed the teenager was 16, not 17 as initially stated, and more than one message was sent.
Sinn Féin referred the matter to the PSNI and social services last September, but no criminal investigation was undertaken.
Sinn Féin leader Mary Lou McDonald has faced criticism for failing to state publicly why Ó Donnghaile had resigned when she issued a statement last December praising his contribution and wishing him well for the future.
Amid growing pressure, First Minister Michelle O’Neill is expected to face questions from a Stormont scrutiny committee next week.
In an email to DUP members on Saturday, Mr Robinson said: “This week has shown us that for Sinn Féin, it’s the same old story of half-truths and cover-ups.
“The serious nature of the McMonagle scandal, followed by the Ó’Donnghaile scandal, demands that Sinn Féin’s leadership face real scrutiny.
“When our DUP members on the Executive Office Committee question Michelle O’Neill next week, it’s crucial that the First Minister tells the truth.
“The cover-ups are fooling no one.”
Sinn Féin has been contacted for a response.
Meanwhile, Mr Benn was questioned on the wider issue during an appearance in Belfast on Friday night.
He was speaking on BBC Radio 4’s Any Questions, broadcast from the Crescent Arts Centre.
Asked about trust in politics, Mr Benn said the protection of children and vulnerable people must be placed above reputation.
“The first thing I would say is there are victims in both of these cases and our thoughts above all should be with them,” he said.
“Secondly, with one of the cases (McMonagle), the legal proceedings are still live, and it would not of course be appropriate to comment on those.
“When things like this happen, does it damage politics? Well, of course it has an effect – of course it does.
“But I would have thought that by now, all of us, all institutions, would have understood from the historic sex abuse cases that have come to light in the last 10, 20, 30 years about what went on in the past, that we would have realised that any attempt to put the reputation of an institution above the protection of children and vulnerable people is wrong, is absolutely wrong.
“And I would say that all political parties, all institutions, all sections of society, have a responsibility to ensure that they put in place measures – proper safeguarding – to protect the interests of those who are children and those who are vulnerable.”
