On Sept. 4, Local 2507 and Local 3621 of District Council 37, along with the E.M.S. Superior Officers Association, filed a complaint with the United States Equal Employment Opportunity Commission, charging that the disparity was also a matter of racial and gender discrimination . Their evidence: Nearly 78 percent of the department’s firefighters and officers are white, and nearly 99 percent of them are male. The F.D.N.Y.’s emergency medical services are far more diverse: About 41 percent are white, 28 percent Hispanic, 21 percent black and 5 percent Asian. More than one in four Emergency Medical Services workers are female.

Labor and civil rights attorneys say a discrimination case would most likely depend on the ability of the E.M.S. unions to show that their jobs are sufficiently similar to that of firefighters to justify similar pay. The case could be strengthened by the department’s history of discriminatory hiring practices. In 2014, the city agreed to give $98 million in back pay and benefits to minority firefighter applicants to settle a class-action suit that argued the F.D.N.Y.’s hiring practices discriminated against racial minorities. In the late 1970s and early 1980s, New York was forced to allow women to become firefighters after Brenda Berkman, an applicant who had been rejected , successfully sued the city.

Mayor Bill de Blasio, whose administration has been in contract negotiations with the unions representing E.M.T.s and paramedics for more than a year now, has dismissed the idea that the disparities are discriminatory, saying the work is simply different than that of firefighters. City officials said giving similar or equal pay to paramedics and E.M.T.s could cost the city some $450 million annually. Cutting down on overtime pay at the Fire Department — which amounted to more than $340 million in fiscal year 2019, which ended in June — may help.

Firefighters do have different jobs. There are far fewer structural fires in New York than there once were, but that doesn’t make the job of fighting them any less hazardous. Firefighters are more likely to die on the job than paramedics and E.M.T.s. Since 2009, nine firefighters have died in the line of duty in New York City, compared with one member of the E.M.S. The figures don’t include firefighters and E.M.S. workers who have died of illnesses related to toxic air in the Sept. 11 attacks.

Yet as the department’s role in emergencies changes, and it becomes a significant provider of medical care, the salary and benefits of its E.M.S. workers must also evolve.

As Ms. Cadet’s experience shows, in addition to a rising workload, paramedics and E.M.T.s regularly encounter hazards similar to those faced by the police and firefighters . A 2013 University of Maryland study, using data from the Department of Labor, found that the injury rate of E.M.T.s and paramedics is three times higher than the national average for the general population . In March, James Booth, the E.M.S. chief at the time, said during testimony before a City Council committee that assaults on paramedics and E.M.T.s by the public had increased by nearly 50 percent between 2015 and 2018, from 79 to 117 .