Club members have been angered following a failed attempt to bring the tapestry to Hawick, regarded by many as the principal centre for textiles in the Borders. And they have made their strong feelings known in correspondence with Mr Parker who is championing the Tweedbank option.
Local MSP John Lamont has also been dragged into the dispute with accusations of his "lack of action" in pressing Hawick's case.
In a letter to Mr Parker, Derick Tait, the Callants Club president, wrote: "Your attempt to shift the blame for inaction on to John Lamont is quite shameful and does you no credit."
Mr Tait claims Hawick is the ideal location for the Borders to capitalise on the Tapestry. He adds: "The late President of the United States, John F Kennedy had a saying - 'An error doesn't become a mistake until you refuse to correct it'.
"Quite obviously an error has been made in choosing Tweedbank as a location for the tapestry. By recognising the real worthiness of Hawick as the best location you can avoid your error becoming a mistake."
Hawick was discarded as a potential host town for the tapestry because there was said to be no suitable building capable of housing it, a conclusion hotly disputed by the Callants Club.
But Mr Tait and his colleagues have vowed to fight on. Their letter to the council leader concludes: " We would urge you to heed the advice given and recognise that if the tapestry is to come to the Borders, then Hawick has the strongest case, and as such is the obvious location."
The council has commissioned a £40,000 feasibility study into the Tweedbank proposals, and the findings are due to be considered later this year. But an announcement last month - credited to First Minister Alex Salmond - suggesting the project would be developed at the southern end of the Borders Railway, took many people by surprise.
It has since been confirmed that the Scottish Government will become a funding partner along with SBC and any external sponsorship that can be attracted for the Tweedbank venture. It means the council will have to commit money to the scheme if it goes ahead.
In a robust rebuttal of Mr Tait's claims, Councillor Parker writes: "I reject entirely your comment that my actions are shameful over the issue of John Lamont MSP's inaction. From everything I have been told, Mr Lamont called the meeting [in Hawick], agreed to follow up one potential site but never made contact again with the two other representatives who were at the meeting.
"From September 2013, John Lamont was well aware the Trustees [of the Tapestry] were seeking a home for the tapestry and it is a reasonable proposition to question whether Mr Lamont could have, and should have done more if he truly wanted the Tapestry located in Hawick".
Mr Parker also explains the recent announcement concerning the project at Tweedbank is still subject to approval by the elected members, with the study, business case and other elements likely to be considered in November.
He adds: "As the leader of Scottish Borders Council it is my role to do what is best for the whole of the Scottish Borders. The fact that we may have secured the Great Tapestry of Scotland for our region, at the expense of other parts of Scotland, must surely be welcomed."
Mr Parker warns that the Trustees have a number of other potential sites that are attractive to them outwith the Scottish Borders that they have indicated they will take forward in the event that Tweedbank is not secured.