The bishops' objections are recorded in letters sent to the library last May, at the time it reversed previous policy and reopened public access to copies of the registers.
Correspondence also shows that the library decided not to publicise the decision so as not to draw any "unnecessary attention".
The library's decision removed a significant source of income for some dioceses which charge fees for access to parish registers.
The Diocese of Cloyne, for example, charges a €25 per family fee for a search of its registers, plus €3 per entry found.
In his letter, the Archbishop of Cashel and Emly, Dr Dermot Clifford, said he was "totally taken aback" at the library's "unilateral" decision to grant access to "allcomers" to the registers.
Dr Clifford said previous arrangements were founded on the "clear understanding" that copyright to the records were the property of the parishes and diocese.
No permission had been given to the library to grant access to the microfilm records.
Access to the registers - fundamental research tools for genealogists tracing family histories - was closed in 1992 after Dr Clifford claimed copyright.
Since then, data from the records could only be obtained by paying for a search at the diocese's heritage centre in Co Tipperary.
In his letter, Dr Clifford reminded the library of the work of the Tipperary centre and urged it to ensure no one was granted access to the library's microfilm records. He continued: "I am reminded in this context of the famous judgment of the High King Diarmuid Mc Cearrbheoil in the case between Finian of Dromin and Colum Cille regarding a copy the latter made of the Psalter which belonged to Finian: 'To every cow its calf, to every book its copy'."
The Bishop of Kerry, Dr Bill Murphy, also wrote to the library to express surprise at the decision and state that no permission had been given to the library to grant access to the records.
However, the director of the library, Aongus Ó hAonghusa, wrote back to both bishops reaffirming the decision and stating that it was made with the benefit of legal advice.
In 1994, 1996 and 2001, the library received separate legal advice indicating no reason why it should not provide public access to the microfilmed records.
In 2006, library chairman Gerry Danaher SC prepared a memorandum for the board in which he pointed out that the vast majority of bishops allowed open access.
The restrictions remained in place in the case of some dioceses "on foot" of the requirement of some members of the hierarchy.
In spite of Mr Danaher's advice, the policy was not changed until this year, following pressure from the Association of Professional Genealogists in Ireland.
The library's collection of microfilms covers the surviving pre-1881 baptismal and marriage records of almost all Catholic parishes throughout Ireland.
+++++++++++++++++++++++++++++++++++++++++++++++++++
No responsibility or liability shall attach itself to either myself or to the blogspot ‘Clerical Whispers’ for any or all of the articles placed here.
The placing of an article hereupon does not necessarily imply that I agree or accept the contents of the article as being necessarily factual in theology, dogma or otherwise.
Sacerdos