Corporations (in their many and varied entities) are creations of Government; they do not have Rights -- they are granted Powers from their creator governments. Powers can impede or restrict Rights under limited and specified circumstance, but cannot abrogate those Rights.
You have a Right to fly anywhere in this country -- if you wish to do so on a commercial airline you are subject to search at any point during that trip. Should you wish to not be subject to those Powers of search, then you are free to fly yourself or hire a private plane.
While similar in application and intent, Powers and Rights are not the same; they are neither exclusive or inclusive with respect to each other.
That's at issue here. Does the parking lot owner have the power to search my car parked at the airport lot at the beginning of the trip?