There have been calls for both Norma Foley and Simon Harris to come before the Dáil and explain how they will rectify the Leaving Cert error.
Opposition TDs say the current "omnishambles" with the calculated grades process has highlighted an ongoing issue with transparency in the department.
Sinn Féin's education spokesman Donnchadh Ó Laoghaire said the news has come on top of a "brutal year" for students.
"It's unacceptable. It has been a brutal year for Leaving Cert students and this is heaping more uncertainty onto that. I feel sorry for students looking at this now and not knowing whether they've been affected," he said.
"What needs to happen is for the minister (Foley) to come before the Dáil as early as possible."
He said that hearing should take place as early as tomorrow, questioning how the minister could find out a week ago and not reveal the news until Wednesday afternoon.
"The lack of transparency is shocking, and we need to hear how this has been dealt with," he said.
Gary Gannon, the Social Democrats spokesperson for education, likewise, demanded an explanation as to why the Government took a full week to confirm that two errors had been identified.
"It simply beggars belief that the Government has known about these errors for a week but kept students in the dark as it scrambled to deal with the crisis," he said.
“The minister needs to extend the deadline for students wishing to apply to sit the traditional Leaving Certificate, which is supposed to be this Friday. In addition, round three CAO offers, which are due tomorrow, need to be postponed until there is further clarity.
"It is imperative that the minister, the CAO and institutions outline quickly what options students have."
Labour's Aodhán Ó Ríordáin said he was astounded by the matter, calling it, "yet another foul-up," while his colleague and spokesperson for higher education senator Anne Hoey said said Mr Harris needs to "clear his schedule" to stand for questioning in the Dáil and Seanad.
"The minister also needs to clarify whether he will make a commitment that for those students who should have been offered a higher place on a competitive course such as medicine, nursing or primary school teaching that a place will be held for them next year if they are unable to enter the course this year," she said.
Aontú Leader Peadar Tóibín noted: "In the same week that a Fianna Fáil senator sought to call in the army to be used against Galway students, they are again at the wrong end of the government’s education omnishambles”.