In a letter to congressional leaders, the White House and Treasury, Parker and the unions warned "nearly 20,000 American Airlines team members are facing furloughs in just two short weeks, and several markets in our domestic network are at risk of significant reductions in air service."
The letter implored the leaders "to find a way to work together to reach agreement on a COVID-19 relief bill that includes a clean extension" of payroll support.
White House press secretary Kayleigh McEnany did not offer details on what actions the administration might take to help airlines.
"The president has mentioned previously that he wanted to take a good look at airlines and help where he can. Of course, we need Congress in a lot of this," she said.
The letter added: "Our conversations with each of you suggest a deal is within reach, and there is no time to waste."
Last month, American said it planned to suspend flights to 15 smaller U.S. airports in October as travel demand remains low due to the coronavirus pandemic.
Congress has been weighing for weeks whether to grant U.S. airlines another $25 billion in payroll assistance that would keep tens of thousands of airline workers on the job for another six months and extend minimum service requirements.
American said it will cancel just over 700 flights in October to and from those 15 airports.
Major U.S. airlines that received portions of $25 billion in payroll assistance were required to maintain minimum flights through Sept. 30, but the government could opt to extend those requirements. American received $5.8 billion in payroll assistance.
By David Shepardson and Tracy Rucinski