When the Cabot School opened in Newtonville in September 1929, the stock market was still a month away from its spectacular fall. The brick Tudor Revival building rose three stories, topped by a cupola.

And looking out with a benevolent, if slightly weary grin, was a sculpture of Humpty Dumpty sitting on a school wall.

Now that historic facade — the nursery rhyme hero included — perches at the center of a debate over how to renovate and expand the outdated elementary school.

In April, the Cabot School Building Committee approved a design that would place a new gymnasium in front of the school, blocking the view from nearby Cabot Park of Humpty Dumpty and other parts of the facade.

But a month later, the group voted to “pause’’ the process so architects could look into the possibility of using newly acquired property to reposition the gym and keep the view intact.

A new option is now on the table, with a vote scheduled for Thursday, that would move the gym to the north side of the school.

“I had reservations about the original plan for a couple of reasons,’’ said City Councilor Emily Norton, who represents Newtonville and serves on the building committee.

“First, I didn’t like that the placement of the gym on the southeast corner of the school obstructed the historical facade that we had voted to preserve.

“Second, I knew that as a stand-alone addition, the gym would cost more in heating and cooling every year, at a time when the city is working to increase energy efficiency.’’

The other two City Council members who represent Newtonville, Susan Albright and Jake Auchincloss, said they also support taking a second look at the options before moving forward.

But a group of Cabot parents see things differently.

Calling themselves Safe New Cabot, they are crying foul on a process they say is putting aesthetics over function in the selection of a final plan.

They say the vote to “pause’’ the process was ill-advised and unnecessary after a plan that satisfied the educational needs of their children had already been approved.

The first plan “prioritizes safety and the educational program — which we should strive to optimize, not simply ‘fit in,’’’ read a statement sent to the Globe by parent Alex Zaroulis on behalf of the group.

“The plans released today do not benefit Cabot students,’’ parent Ben Josephson said after the new option was unveiled. “The educational programming, traffic and public safety, and impact on our neighbors are objectively worse.’’

In addition, the parents argue that the first option was approved after months of vetting and discussion, while the second is scheduled for a vote just nine days after being presented.

“Cabot students deserve better. They deserve the fully vetted plan passed in April,’’ Josephson said.

The Cabot renovation will be the third elementary school project paid for by a debt exclusion voters approved in 2013.

Students moved into the new $37.5 million Angier School in Waban in January, and work is now underway on a new Zervas School.

Students from Zervas are now attending classes at Carr, where Cabot students will go once construction on their school begins.

The exact cost of the Cabot renovations will not be known until later in the process, but part of the funding will come from the Massachusetts School Building Authority.

While the parents’ group has thrown its weight behind the original plan, many others, including people across the city interested in historic preservation, opposed the design, according to Albright, who pushed for the pause in the process.

Albright also questioned whether the original plan would get the necessary approval from the City Council and Newton Historical Commission without fully considering the options available since the city purchased the so-called Potter property next to the school at 23 Parkview Ave.

“The Potter property was never given a fair shake,’’ she said.

School Committee member Diana Fisher Gomberg, who is also a member of the Cabot building committee, agreed.

“While I was satisfied with the plan we had already approved in April, it had become clear that some city councilors required a closer look at the possibility of locating the gym to the north side of the building before they could support the project,’’ she said.

City officials say they regret the pause, but they insist it will not jeopardize state funding or delay the project, which is on track for completion by September 2019.

“We were very concerned that had the City Council voted this down, we would have lost much more time,’’ said Dori Zaleznik, the city’s chief operating officer and a member of the Cabot building committee. “Was the process flawed? Yes, but let’s look and see what this new option offers.’’

The new option would put the entire addition on the North side of the existing Cabot School, with a gymnasium, cafeteria, music, and art rooms all on the first floor, and classrooms on the second and third floors.

A library would also be on the third floor, near the fourth- and fifth-grade classrooms, according to Zaleznik.

There would be direct access from the gym and cafeteria to outdoor play space, she said, as well as room on the Cabot site to move the play structures, which would be put on Cabot Park in the first plan, back onto school property.

“There was sense from the committee that this is a viable second option to explore on June 2,’’ she said.

But parents from Safe New Cabot have their doubts, saying the new option will also have critics.

Putting the entire addition on the north end of the existing school “will burden the direct abutters on Parkview and Bridges avenues, disrupt the flow of student drop-off and pick-up that residents have worked hard to ameliorate in the approved plan, and dwarf the current structure with an appendage twice its size,’’ the statement from Zaroulis read. “There is no value to future students to move to an alternative plan at this point.’’

Ellen Ishkanian can be reached at eishkanian@ gmail.com.