Cllr David Robson branded the promotion of planning boss Alan Menzies to deputy chief executive as “unfair” at a time of austerity.
The UKIP member, for Bridlington South Ward, also criticised a pay increase for Mr Menzies.
The current director of planning and regeneration’s yearly take-home will increase to Â£122,000.
Cllr Robson said: “At what point that decision was taken seems unclear, but it must have been decided amongst the four committee members during the interview process, otherwise it would have been included on the authorised mandate presented to council.
“Furthermore, if a proper decision had been taken after discussion amongst the councillors, the role should have been advertised internally, allowing other senior officers to apply for it, who may not have wished to apply for the full role of chief executive.”
The role of deputy chief executive is the first in the council’s history.
Mr Menzies will answer to Caroline Lacey, who was picked ahead of him to replace outgoing Nigel Pearson as the authority’s new chief executive.
She will take home a salary of Â£158,000.
Cllr Robson’s amendment to refer the matter for further discussion was defeated by a full council vote.
The authority’s leader, Cllr Stephen Parnaby OBE, accused opposed members of “nit-picking and carping” over the issue.
He said the appointment of a designated deputy chief executive was the “right fit” for the authority’s senior management team – and that role was common at other local authorities.
He added while Mr Menzies would receive a salary increase, that the council would make an overall saving through a reduction in the number of senior directors at the authority over recent years.
Cllr Robson told the Free Press: “He did not attach any importance that the appointment sub committee, had gone beyond their remit, given to them by full council, which made no mention of anything else other than to short list, interview and appoint a replacement for the retiring chief executive.
“The councillors had absolutely no prior knowledge and therefore never agreed to making a role of a deputy chief executive permanent.
“It is astonishing that Councillor Parnaby dismisses the fact that the procedure and convention of the council can be ignored by him and dismisses everyone else who pulls him up with the insult that they are ‘nit-picking and carping’.