He took over America West Airlines just 10 days before the terrorist attacks of Sept. 11, 2001, and steered the company through a brush with bankruptcy. When America West merged with US Airways in 2005, Mr. Parker — who was running the smaller of the two companies — took over the enlarged group.

He pulled off a similar trick eight years later. With consolidation of the industry in full swing, American Airlines merged with US Airways. Once again, Mr. Parker, running the smaller airline, emerged as the chief executive of the new parent company.

But this year, Mr. Parker has faced a new sort of problem. With the grounding of Boeing’s 737 Max after two deadly crashes, American Airlines became one of three American carriers to take planes out of service. The delays have persisted longer than expected as Boeing has struggled to get the plane recertified by the Federal Aviation Administration, forcing American to cancel tens of thousands of flights and lose millions of dollars in sales.

Patience is running thin. When American reported quarterly earnings last month, Mr. Parker emphasized that he expected Boeing to compensate his company in full, and stressed that airlines were struggling with the Max still grounded.

This interview was condensed and edited for clarity. It was conducted at American Airlines’ offices in New York City in late October, the week before Boeing executives testified before Congress.

After the first 737 Max crash, of Lion Air Flight 610, what did you do to make sure that this wasn’t a problem with the Max?

I don’t recall any real sense that it was an issue with the aircraft itself. But we didn’t know, obviously. Our team was following it, and we rely on the Federal Aviation Administration to represent the safety interests of the industry. Any time there’s an incident like this, there are immediate efforts to ensure that it doesn’t happen again.

Did American ever consider voluntarily grounding the Max before the second crash?

Not to my knowledge.

What were you hearing from Boeing after the first crash?

Boeing was not contacting me at that point. I’m getting reports on it. But in the time between the first and second accident, none of the reports we were receiving were raising any sort of indication that there was a problem with the airplane.

Our conversations with Boeing were about what was needed to be done to ensure it was safe. What we were being told is that there will be a software fix. The date, I can’t remember it anymore, because they all slipped so much. But that’s where we were.

Then Ethiopian Airlines Flight 302 crashes in March. What was going through your head when you learned that it was another Max?

We need to know what’s happening. We were in regular contact with the F.A.A. at the highest levels to figure out what is the issue, and are our aircraft safe? Every piece of information we were getting at that point led us to believe that our aircraft with our pilots were safe to fly.

Were you comfortable with the F.A.A.’s decision to let the planes continue flying even after international regulators grounded the Max?

Absolutely. If we thought for a second that our airplanes with our pilots were unsafe, we would have grounded them ourselves. If we had any concerns whatsoever, if any of our pilots had any concerns whatsoever, I assure you they wouldn’t take the aircraft up.

What’s your sense about the role the pilots played in each of these two crashes?

I don’t have a view on that.

You’ve been more vocal with your frustration with Boeing in recent weeks.

The F.A.A. has been very forceful in saying, “We will decide whether or not it’s safe.” They’ve been adamant about that, which I think is exactly the right stance to take. They have set certain conditions to make that assessment. Boeing doesn’t appear to be meeting the F.A.A.’s deadlines.

It’s not the F.A.A. changing the condition. It’s the manufacturer either setting too aggressive a view as to when they will have the conditions met, or not being able to deliver what they said they were going to deliver. Whatever the result, the F.A.A. is not changing their requests, but the aircraft still continues to slip in certification.