I wanted to know if my P.O. Can call the airport and ask if John Doe Left the State? Or even bought flight tickets?

Probably not, if you do it.
They might think terrorist tracking or something, and not only tell you nothing, but do a through research on your current and past.
If a P.O. (private investigator right?) is legitimate, they should be able to access those files either online or through the airport themselves. They'd have to know who to call too, most airports don't have an extension for people who have previously bought tickets. But it's worth a shot. If the cops can do it then why not a PI?

PO = Parole Officer

Believe me, they'll find out, one way or another.

With all the security with flying these days, I wouldn't be surprised if there's a way to block you from even getting on the plane to begin with.

I'd like to know why you're skipping out of the state while you're on parole.

Now I feel dumb.
Yea, a parole officer deffinetly. Even if they don't call the airport, they'll notice your missing eventually and then just look at your recent files in their databank and bam they see where you've gone.
Is it just vacation, or are you skipping town? Vacation maybe you could get away with but the other probably not.

It's a Probation officer for a minor crime.

Still if they check in to it, they can find out.