Holyrood's public audit committee has written to the justice secretary saying SPA chairman Andrew Flanagan appeared to have "behaved inappropriately".
The letter follows a series of heated evidence sessions at parliament, which saw one MSP tell police bosses "it's not the Kremlin you're running".
Mr Flanagan insisted the group has made "substantial progress" over 12 months.
MSPs have been conducting an inquiry alongside a separate probe by HM Chief Inspector of Constabulary in Scotland Derek Penman into the governance of the SPA and concerns about transparency.
The letter from the committee, which was addressed to Justice Secretary Michael Matheson and copied to Mr Flanagan, is highly critical of the chairman and the running of the board.
It reads: "We are writing to set out our very serious concerns about the standards of governance at the SPA, following our recent evidence sessions.
"In particular, we consider that the chairman of the SPA board, Mr Andrew Flanagan, would appear to have behaved inappropriately on occasion and in a manner not in keeping with relevant Scottish government guidance.
"We consider this to be unacceptable, particularly in relation to a public body that performs such a vital role."
The committee were particularly critical about a row over meetings being held behind closed doors, which saw one board member, Moi Ali, quit. She told members that she felt she had been bullied, describing her exit from the board as "a really horrendous experience".
The letter notes: "It appears to us that Mr Flanagan treated Ms Ali in an inappropriate manner, to the degree that she felt obliged to resign from the board."
While members acknowledged there would be times when public bodies would have to discuss matters in private, they said the "default position for such an important body is that its committees should meet in public".
And they said it would be "entirely unacceptable" for the SPA to repeat the situation where private meetings were held to discuss governance with no notes being taken, saying there was "a clear need for a culture shift within the organisation so that there are far fewer private meetings".
On the matter of transparency, they said some of the board's "decisions on basic operational matters have been inexplicable", and said SPA board members "should be far more critical in how they question or challenge some of the decisions made by the chair of the board".
In response to the letter, a spokesman for the SPA said: "As board members outlined this week, the SPA is listening to public and civic concerns and has already signalled it is ready to adapt its approach at the next public board meeting.
"In addition, HMICS is currently looking at governance within the SPA and we are working closely with the inspectorate to facilitate that review. Its findings and recommendations will further inform our approach to improved governance and transparency."
During his appearance at Holyrood in April, Mr Flanagan insisted the group had made "substantial" progress over the past year, adding: "I believe I am doing an effective job."
He said: "I think we've made a number of substantial movements within the last 12 months based on the governance review.
"I think we are becoming more effective, I think it is important that we recognise that there is already a high and significant degree of openness through the public board meetings that we have which are second to none in terms of public bodies in Scotland.
"I think we are on a journey. Is it perfect? No, it's not perfect, but I think in the last 12 months or so there have been significant steps forward."
And in a subsequent sitting on Thursday, current board member George Graham said there had been a "changing atmosphere" since Mr Flanagan took over, saying the board was "much more engaged" with "a clearer sense of purpose".
MSPs have voiced "very serious concerns" about how the Scottish Police Authority is run.