Leaders of the pilots' union say they have lost confidence in the ability of Parker and senior executives to lead the airline. Flight attendants picketed Tuesday at company headquarters and three big airports.
The unions are complaining about lower pay and profit sharing than counterparts at Delta Air Lines. Delta said it will pay about $1.1 billion to employees as their share of the company's 2016 profit. American set aside $314 million.
American says it has increased wages and benefits by $3.5 billion since its 2013 merger with US Airways.
Airlines have become hugely profitable in recent years after a string of mergers. Parker has pledged that American will provide industry-leading pay when contracts come up for renegotiation, but that isn't until 2020 for American's pilots. They wanted upgrades after Delta and United pilots got raises last year.
The Allied Pilots Association board unanimously approved a resolution on Monday saying it had lost confidence in Parker. Union President Dan Carey said American has made questionable decisions that have kept the airline behind Delta in customer satisfaction, operations and revenue.
Company spokesman Matt Miller said American shares the union's goal of making the airline a great place to work and is pleased with its progress, so "further public dialogue serves no purpose."
Separately, American flight attendants were picketing Tuesday at the company's headquarters in Fort Worth, Texas, and at airports in Los Angeles, Miami and Charlotte, North Carolina — all busy hubs for American flights.
Union President Bob Ross said that despite record profits, American flight attendants are paid less than at other airlines and are unhappy about frequent computer meltdowns, bad schedules and new uniforms that some employees say cause allergic reactions.
Miller said flight attendants have received average pay increases of 27 percent since the merger. The company said in November that pilot pay had climbed an average of 53 percent in that time.
The vote by the pilots' union board and the picketing by members of the Association of Professional Flight Attendants have no legal effect but symbolize worsening relations between the unions and senior management.
In 2013, Parker successfully courted the unions in his bid to force then-bankrupt American, which had a history of stormy relations with labor including strikes in the 1990s, to merge with his smaller airline. Parker's team replaced the executives who were running American.
Shares of American Airlines Group Inc. fell 84 cents, or 1.8 percent, to close Tuesday at $46.57.
(David Koenig - Associated Press / ABC News)