American Airlines (AAL.O) warned airline passengers may eventually face higher ticket prices if oil prices remain high, prompting carriers to remove seats from the market.
Oil prices have risen around 50 percent compared to the levels seen last year and that is putting pressure on airline profits.
The United States last month agreed a deal with the United Arab Emirates and in January with Qatar to resolve claims from the three largest U.S. carriers that Gulf airlines had received unfair government subsidies.
Parker said he was pleased with the result of the talks but needed to see more from the Gulf carriers before American Airlines could consider partnerships with them.
“We haven’t had enough time to make sure that those resolutions have the effect that we hope for, so we shall see,” he said.
Qatar Airways had plans to buy a stake in American Airlines last year, but reversed that decision, saying an investment did not meet its objectives. American Airlines executives had opposed the share buy.
When asked on Sunday whether he would be open to investments from Gulf airlines further down the line, Parker said American Airlines did not need such transactions. He added, however: “We are open to investments from everybody.”