The company beat Wall Street expectations on the top and bottom line earlier in the morning. The airline reported earnings of $1.76 per share on $10.59 billion in revenue. Analysts estimated EPS of $1.69 and revenues of $10.56 billion.
Parker said that while revenues are down year over year across the airline industry, they're down less for American Airlines than for other companies. He expects revenue to be slightly better during the fourth quarter.
Shares of the airline were trading lower nearly 1 percent Thursday afternoon.
He added that the company's fourth-quarter issue is a "cost issue," because of investments the company has made in its team, he added.
"Primarily, things like profit-sharing, things like increases in pay as we've gotten union contracts done, those are behind us now, and that's great because we've done a lot for our team," he said.
When asked about rising jet fuel prices affecting airline shares and fares, Parker said there is now more of an ability to raise fares because the supply of jet fuel is slowing down, benefiting the company's stock and its investors.
The U.S. Department of Transportation recently came out with new rules regarding the transparency of airfare advertisement. New regulations include the end of airlines' selective posting of airfares and things like refunds for delayed bags.
On that subject, Parker said he looks forward to the day the government "doesn't feel like they need to take care of our customers, because we're doing it."
Regarding the election, the airline's chief executive said it's "gotten to a point where it doesn't feel like it's good for business." He said those in the business sector should be more involved in being "socially progressive" because of their customers and all the people who work for them.
"I think we in the business community need to get to work on this," he said. "There's nothing out there that feels like it's going in the right direction for U.S. Congress."