Part of the problem isThe answer depends on how the word weight is defined, which then dictates the meaning of the word weightless. Of course, there can be, and are, multiple definitions. As many (if not most) of the posters here are experts (whatever that may mean) in physics or astronomy, they of course have a greater intellectual library from which to draw. Some of us have specific audiences in mind when we answer questions here. I argue that audience must include introductory students who may, indeed certainly, look here for answers.
By definition 1 (which was the definition presented to me in my undergraduate and graduate courses at two universities), weight is the nickname for the gravitational force on an object due to Earth. No matter where the object is, it has a well defined weight. During a parabolic flight, the passengers experience essentially the same gravitational attraction to Earth that they experience standing on the ground. The attraction diminishes by only about 10% in low Earth orbit (e.g. aboard the ISS).
Some introductory textbooks incorrectly define weight as the reading on a spring scale when the object is on the scale, and indeed this is a correct operational definition of weight, unless you expect numerical agreement with definition 1 all the time. The scale doesn't measure the object's gravitational weight if you're accelerating; the number depends on the object's state of motion. In general, the spring scale only measures the magnitude of the contact force pressing against the object, and it just so happens that if the object (and the scale) are not accelerating vertically, its weight is numerically equal to the contact force on you from the scale. This approach should not be used, especially in introductory courses because it's dependence on acceleration is confusing to non-experts.
A potential consequence of definition 2 is that without a contact force, one must then deal with questions about how and why things float inside spacecraft in low Earth orbit, and this issue is inevitable among even sophisticated students. Yes, the experts among us know that there's no floating involved and that the craft and occupants are merely free falling.
With respect to the sources using definition 1, NASA creates problems for students and interested laypersons by propagating the term weightless. For sources using definition 2, that particular complication doesn't exist, but another arises from the need to involve acceleration. Which one is better is not dictated by the most frequently used one, as that constitutes a fallacious appeal to the populous and perhaps also a fallacious appeal to authority. Which one is better should be dictated by which causes the less confusion and fewest misconceptions, a criterion easily measurable in the classroom.
Regardless of which definition of weight one uses, I hope we can agree that if the passengers and aircraft are all in a state of free fall, there is nevertheless a gravitational force due to Earth on them, and that this force exists no matter what, and that no contact force (aside from casually bumping into the aircraft's interior or another passenger) exists between aircraft and passenger.