For years, mounting tension between parents and the principal at New Explorations Into Science, Technology and Math High School, one of the most sought-after public schools in the city, has been a hot topic of conversation, with parents quietly -- and sometimes not so quietly -- critiquing the administrator's communication skills, her hiring acumen, even her “quirky” demeanor.

But the discord reached a discomfiting peak this week when more than 500 parents and faculty members signed an online petition that chides the principal, Olga Livanis, as being overly punitive in this year’s rating of several well-liked teachers, for having not shared important budgetary information with parents, and for the tone she set at the school, which has 1,600 students in kindergarten through 12th grade and is on the Lower East Side.

The petition -- a laundry list of complaints or an attempted coup d’état, depending on whom you spoke with this week -- accuses Dr. Livanis, who has been the principal since 2006, of giving several well-liked staff members "unsatisfactory” ratings in their annual evaluations, a move that could jeopardize their potential to get a raise and positions teaching summer school and coaching after-school activities, not to mention get another job.

The petition, which implored Dr. Livanis to “please work with us to address the concerns stated above,” also takes issue with the decision to also make the school’s parent coordinator the pupil accounting secretary, which parents say has left her unable to do her job full time. It characterizes the School Leadership Team as “dysfunctional.” And it objects to a staffing decision this past school year that left the school without money for dance, chess and technology classes before the school’s PTA stepped in to cover the costs.

Wrote one parent, Zhenia Stadnik, who signed the petition, which was made public earlier this week: "I wholeheartedly support this petition." She added: "And the issue of clear communication is of paramount importance.”

“Thank you for hearing us,” wrote another.

Dr. Livanis is traveling out of the country, but responded by e-mail to say that she would be meeting with the PTA leadership in the coming weeks to discuss the concerns.

A Department of Education spokeswoman, Erin Hughes, said officials were aware of the complaints and were “working closely with staff at the school to address them.”

Parents at the school, which is commonly known as NEST+m, said they were particularly concerned and puzzled by the poor teacher grades because they were doled out to well-liked staff members, including a popular middle-school math teacher, a favorite physical education teacher and a third-grade teacher who has been at the school for several years.

One mother who signed the petition called one of the teachers who received the rating “a marvelous role model, mentor and teacher for our students.”

About the ratings, the mother wrote: “I think there is something wrong with our ratings, not with our teachers.”

Other parents gushed about the enthusiasm and skill set of poorly rated teachers and called the ratings of cherished teachers “disheartening.”

The petition was followed by an anonymous letter sent to officials in the Department of Education that took further aim at Dr. Livanis and characterized the culture of the school as “one of fear and intimidation.”

The letter echoed the concerns of parents like Rachel Leinweber, who says the school has seen unprecedented turnover in the assistant principal's position in recent years and called Dr. Livanis “antagonistic.”

“It’s mind-numbing,” said Ms. Leinweber, who has written several complaints to the Department of Education.

Ms. Livanis has been principal at NEST+m for six years. Before that she was assistant principal of physics and science at Stuyvesant High School, where she had taught chemistry, biology and general science.

Parents said Wednesday that they were not all seeking the same outcome. Some wanted Dr. Livanis to step down, or even fired. Others said they simply wanted her to become more responsive. And others said they wanted her to know, “We’re watching you.”

“Our attention is not to lambaste the school, but to let her know our teachers and our kids need more support,” said Michelle Luhan, a parent of a seventh grader and a fifth grader.

This is not the first time parents at NEST+m have had issues with the principal. The previous principal, Celenia Chevere, resigned after a dispute with the Department of Education over the city's plans to put a charter school in the building.

And Nest+m is not the only specialized school where there has been friction involving its leadership in recent years. Valerie Reidy, the principal at the Bronx High School of Science, another school with a demanding entrance exam, has also been accused of ineffectively communicating with parents and faculty members and of awkwardly addressing faculty issues.

Ms. Reidy has defended herself openly, and in an article in New York magazine last year claimed her detractors were simply resistant to much-needed change at the school.