You're right, and your parent is not, for two reasons.
Firstly, gravitational radiation is observed to obey the classical massless wave function, just as in the large-number-of-photons-limit light obeys the classical massless wave function.
The second quantization [1] of each such massless wave function leads to a massless gauge boson of spin-2 and spin-1/2 respectively: the graviton and the photon. There is excellent experimental and observational support for this approach as an effective field theory -- as one takes the energies of the particles in either field (in isolation) higher, one runs into theoretical questions that have not been resolved.
However, this second-quantization approach conflicts with the approach taken in the Standard Model, which defines a massless gauge boson (also called a photon) and is silent about the quantum content of gravitation [2]. The photon is massless because it moves at "c", and vice-versa. For a Standard Model graviton to be defined, it must also be massless, or light must not always move at "c", leading to photons of different energies moving at different speeds (in vacuum) relative to an observer of those energies. This conflicts with experiment.
The "bigravity" [3] family of gravitational theories probes this variable-speed-of-light problem, and are amenable to study under the Parameterized Post-Newtonian Formalism with results that conflict with evidence [4]. In General Relativity, distant emitters of electromagnetic radiation and distant emitters of gravitational radiation must line up in the sky barring intervening matter that interacts with light. This is in fact what we observe in the Ligo/Virgo era, and since the Mercury MESSENGER experiments. In this case it's because our universe is Lorentzian, having 3 dimensions of space and 1 of time. In Lorentzian universes in General Relativity there is one type of geodesic ("lightlike" [5]) along which massless objects may move, and that geodesic is forbidden to massive objects.
The idea of "massive" gravitational radiation is a theoretical curiosity that is undermined by new evidence gathered practically daily (e.g. in the results of sky searches for supernova and binary eclipses (and other multibody eclipses) by e.g. ASAS-SN : http://www.astronomy.ohio-state.edu/asassn/index.shtml ).
Secondly, gravitational radiation can be included in exact vacuum solutions to the Einstein Field Equation of General Relativity, and this is grad student textbook and lecture note material. The notable feature of vacuum solutions is that the stress-energy tensor T_{\mu\nu} is defined to be zero everywhere in the spacetime. That one introduces "test probes" into the spacetime to see how they move under the influence of gravitation does not change this crucial feature.
For the most part, one should take "energy" and "momentum" as referring to coordinate-system-dependent components of the stress-energy tensor. If we write down the stress-energy tensor as a 4x4 matrix, labelling 0..3 on the rows and columns, with a different matrix at every point, then we can think about the matrix at one point pretty straightforwardly as showing the flux of momentum into the point from each dimension of space or time, and the flux of momentum out of the point along each dimension of space or time. One conventionally takes energy or mass-energy as the time-time component: momentum that comes to this spacetime point from the past and leaves this spacetime point for the future, the spatial coordinates being constant. (We'd write this as T_{00} != 0. Compare the totally inelastic absorbtion of a photon from "the left" (spacetime direction 1) that we'd write as T_{10} != 0 because the momentum stays at the same spatial coordinates going into the future.) But in a vacuum, T is everywhere zero, so there is no energy, stress-energy, energy-momentum, or however you want to label the nonzeros (generally this depends on how one slices up the tensor into components).
In a vacuum solution, it is difficult (and usually meaningless) to talk about the "energy" of gravitational radiation because there is no matter to feel it, and it usually has to be defined on some surface at infinity; this is because the only nonzeros are in the Einstein tensor.
Alternatively, one can impose a notional "box", to try to tease out the wording of your parent comment. In a Lorentzian universe, this can be done with pseudotensors, but these are fragile to changes of systems of coordinates (which is a "bad code smell" in relativity). Essentially one draws a boundary around a region of spacetime and counts the contents of the pseudotensor and the stress-energy tensor on either side of the region's boundary. This is perfectly reasonable in practical astrophysical applications of General Relativity, but is not a good foundation on which to build an argument that "a box of full of [gravitational waves and nothing else]" is non-empty. It is more in line with General Relativity to study a box of electrically neutral gas immersed in an otherwise-vacuum spacetime that contains gravitational waves, and study the evolution of the stress-energy of that gas. Yes, the gas's equation of motion depends on the gravitational waves, and could in principle be heated or cooled by the interaction with the gravitational waves, but when you step back what you are seeing is the behaviour of the sources (the stress-energy, the gas in the box) telling an otherwise vacuum spacetime how to curve. The objection to thinking of gravitational waves as having some peculiar energy-momentum is mostly that it distracts one from that fundamental point, and the follow-on that when "curvature tells matter how to move" the moving matter backreacts on the curvature. One gets lost very quickly in realistic general-relativistic spacetimes when one loses sight of matter as the background-independent sources of curvature.)
So, in summary, using conventional notions of mass, you are right that massive gravitational waves are unphysical (except maaaaaaaaaybe in the extremely early universe, but that's speculation that hasn't (yet) been (wholly) eliminated by evidence). Additionally, gravitational waves are insubstantial -- they can be represented in a vacuum, which is by definition devoid of any substance -- so it is perfectly fine (and usual practice, in my experience) to call a region full of gravitational waves "empty space".
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[1] https://en.wikipedia.org/wiki/Second_quantization
[2] The Standard Model fields are all Lorentz-covariant and so work everywhere that the radius of curvature is much larger than the particle wavelength. That is pretty much everywhere in the universe except near very small black holes (as yet unobserved), deep inside bigger black holes (possibly unobservable in principle), and very near the hottest densest phase of or universe (not yet directly observed, but plenty of indirect evidence). The coupling of the Standard Model to gravitation is more than good enough to do accurate and precise high-energy astrophysics (the spectra of supernovae and blazars; the equation of state for neutron stars) for the time being.
[3] https://en.wikipedia.org/wiki/Bimetric_gravity
[4] https://en.wikipedia.org/wiki/Alternatives_to_general_relati... in the table "Bimetric" for three examples (Rosen, Rastall, Lightman-Lee). The nonzeros in the \alpha_1 parameter are fatal to these theories as that parameter is very highly constrained by direct experiment involving human artifacts and around other bodies in the solar system; the \alpha_2 parameter is highly constrained to zero by observations of millisecond pulsars, which casts serious doubt on the non-zeroes in that column. \gamma is also constrained by experiment within the solar system. One has to "wash out" the effect of bigravity by making the decoupling or vanishing of the second metric happen very very near the big bang (so that light behaves entirely masslessly at all energies when the distortions in the cosmic microwave background develop). A "washing out" can still have effects early in cosmic inflation, so these theories are not dead, just that they predict smaller and smaller differences from standard single-metric General Relativity.
[5] https://en.wikipedia.org/wiki/Causal_structure#Tangent_vecto...
[6] http://aether.lbl.gov/www/classes/p139/homework/hw12.pdf