The Social Democrats say they have tabled a Dáil motion for a public inquiry
Garda whistleblower Sergeant Maurice McCabe has called for a full public inquiry into allegations of a smear campaign against him.
Last week, the Government announced the formation of a Commission of Investigation into the ongoing scandal.
It also emerged last week that that the child and family agency Tusla mistakenly created a file containing the false allegations about Sgt McCabe as a result of a “clerical error".
However, there are increasing calls for a full public inquiry into the controversy.
In a statement released this evening, Maurice and Lorraine McCabe say their experience of the O'Higgins Commission of Investigation "is too fresh in our minds to allow for a repetition".
They argue: "The public has never been made aware that, throughout the proceedings before that Commission, Maurice, at the hand of the legal team representing the current Commissioner, was cast in the role of culprit and/or defendant, and as a person making those complaints in bad faith and without cause.
"Because the 2004 Act prohibits under pain of criminal law the publication of the actual evidence tendered to such Commissions, the public has little or no appreciation of what was done, and attempted to be done, to Maurice in the course of its hearing."
The statement also says that the entire transcript of the previous Commission is in the possession of the Justice Minister.
The Social Democrats, meanwhile, say they have tabled a Dáil motion for a public inquiry into the McCabe controversy.
The motion also calls for Garda Commissioner Nóirín O'Sullivan to step aside for the duration of any inquiry and "replaced by an international senior police officer, in a caretaker capacity".
Social Democrats co-leaders Roísín Shortall and Catherine Murphy said: "The McCabe family have been through a shocking and appalling time. The only way to ensure that the McCabe family get the truth and justice they deserve is to establish a full Public Inquiry.
"Such an Inquiry will also send a clear message to all whistle-blowers that they will be safe in Ireland."
For the planned Charleton Commission, the McCabes say that they have "consistently submitted that any further into these matters must be a public inquiry."
They say they want answers to a number of questions related to the false allegation of child abuse, including "who was/were the Gardaí who made/received phone calls" over the allegation.
They also are also asking "was any of the foregoing Garda activitiy notified formally or informally to senior Gardaí at Commissioner level, and if not why not?" and "was any decision made not to inform Maurice of the making of the 2013 allegation, and if so why and by whom?"
The McCabes said: "We know that the present Commisioner of An Garda Síochana has calmed in public to be supportive of us while seeking in private to discredit Maurice McCabe before the O'Higgins Commission. We have witnessed with growing disbelief her denials of involvement in discrediting Maurice.
"If the foregoing questions are answered truthfully and in public now, the public will be in a position to know who to believe."
Speaking to Newstalk Drive, the Irish Examiner's Mick Clifford said: "What was attempted in their view, and in the view of the evidence we've seen behind the closed doors of the O'Higgins Commission... I don't think anyone who is subjected to that would be willing to go through it again."
Fianna Fáil Justice Spokesperson Jim O'Callaghan said: "We fully support [the McCabe family's] call for a public inquiry to be conducted into this matter. We also believe a public inquiry will benefit An Garda Síochána and the general public."
Labour leader Brendan Howlin said: "In their pursuit of truth and justice, the McCabe family wishes for a public inquiry must be upheld."
Sinn Féin is also backing the McCabes' call for a full public inquiry.
The party is also still insisting that a general election is needed and that the Garda Commissioner must step aside.
Deputy leader Mary Lou McDonald says a Commission of Investigation is no longer viable when the person at the centre of the matter has 'no confidence' in such a Commission.
The McCabes' statement - which can be read in full here - comes as Children's Minister Katherine Zappone ordered an independent statutory investigation into how Tusla manages allegations of child abuse.