The Welsh club described the LMA's criticism of the timing of the release of the dossier regarding Mackay and former head of recruitment Iain Moody to the Football Association as "preposterous".
They also said the 'friendly banter' defence of Mackay's text messages was "reprehensible"
"A manager's behaviour should demonstrate to players and other employees under his control that discrimination in any form is unacceptable," said a Cardiff statement released through the club's lawyers Mishcon de Reya.
"We therefore find it entirely reprehensible that the LMA should itself put out a statement which seeks to dismiss deeply offensive racist comments as 'friendly banter'.
"If that is the view held by the LMA, as appears from its statement, we consider that Richard Bevan's position is untenable and we call for his resignation."
Cardiff say the LMA had known about the existence of the offensive messages concerning Moody and Mackay for months and had chosen not to deal with the issue
They say the LMA was therefore "complicit in the attempt to conceal these messages" and also insisted there were "many more" messages from Mackay than the two he admitted to sending in the statement released on Thursday
The statement added: "We had not intended to comment on matters which have been reported in the press over the last two days in relation to Malky Mackay and Iain Moody
"However, following the statement released by the LMA last night we feel compelled to respond
"The LMA is the representative voice of managers, and whilst we understand it seeks to act in the best interests of its members, one of its major aims is to 'encourage honourable practice, conduct and courtesy in all professional activity'
"Regrettably, we feel that the LMA has done no such thing in its representation of Mr Mackay and Mr Moody
"The LMA were aware of the allegations from the morning of the search order, and their lawyers acted for Mr Mackay and Mr Moody throughout the legal process
"That the LMA has sought to criticise the club for the timing of the report to the FA is preposterous, because the offensive communications have been in the knowledge and possession of the LMA for many months
"When the messages came to light, over three months ago, the club strongly encouraged and advised Mr Moody and Mr Mackay to deal with the issue directly with the FA
"It was made clear to them, and their LMA-appointed lawyers, that the nature of the communications meant the club was under a duty to report their findings to the FA if they did not take appropriate action themselves
"Nevertheless, with the backing of the LMA, Mr Moody and Mr Mackay chose to do nothing
The LMA were therefore complicit in the attempt to conceal these messages (of which there were many more than the two isolated texts acknowledged by Mr Mackay)
"We also point out that the LMA members have a 'Code of Conduct' incorporated into their managerial contracts
One of the requirements is 'A manager shall not use racist or other discriminatory language'."
Source : PA
Source: PA