Revd George Whyte, clerk to the Church of Scotland's Edinburgh Presbytery, is warning such a decision would "knock the life out of" congregations.
In a statement which Premier has confirmed, he told the Edinburgh Evening News: "Our congregations see a bleak future for themselves if this goes ahead. Churches will close and what they do will be lost.
"There's a genuine fear the council are blundering into this without understanding what will happen."
In a meeting next Tuesday, councillors on the Transport and Environment Committee at Edinburgh City Council are due to decide on one of three options for the future of parking in the Scottish capital.
In the process of finalising a Parking Action Plan, they have already agreed in principle to extending parking controls to Sunday but exactly how this will work in practise is yet to be decided.
The three options on the table include plans to only have charges between 1pm and 6.30pm, while another proposal would see charges operating between 8.30am and 6.30pm.
Transport Convener Councillor Lesley Hinds said: "We've already agreed that we need to introduce some parking controls on Sundays, as the status quo is simply not sustainable now that Edinburgh, like so many other places, has become much more of a 24/7 city in terms of socialising and shopping habits.
"What we need to do is to achieve a balance between alleviating uncontrolled parking and increasing turnover for businesses and making travel easier for residents and visitors."
"It will be for Committee to decide, based on how the different options fit with consultation feedback and so on; once an option's been agreed, we can finalise the Parking Action Plan and get ready to put it into action."
Councillor Lesley Hinds is not expecting proposals for morning charges to be approved and 80-per-cent of residents opposed parking controls being brought in on Sundays and evenings, according to the Edinburgh Evening News.
It is estimated introducing parking charges on Sundays in Edinburgh could bring in as much as a £595,000-boost for Edinburgh City Council each year.