Might be too late, I'm not sure. Perhaps you can start another bounty and award it. No big deal either way, it was instructive for me to work through the problem. I guess you already figured it out the problem before my answer?

I can do that, but I am not familiar enough with these functional analysis issues to do much more than cite the sources the give in your comment.

You are certainly correct that it does not address the $H^2$ issue. My assumption (which appears to be erroneous) was that the observation most troubling OP was the absence of overlap with the eigenbasis, and that resolution of this problem would be sufficient for resolving his question, "Is the parabolic state valid?". I presumed OP would accept that the conflicting results for $\langle H^2 \rangle$ is due to some mathematical technicality, but the lack of overlap is a more major conceptual conflict requiring resolution.

Good point @Tobias, once I saw the error in the set-up (all the eigenstates are odd functions w.r.t. reflection about the well center), I did not keep reading. However, OP's question about the "validity" of the state seems independent of its energy expectation value $\langle H \rangle$.

The image of the double-cover mapping $G$ to $SO(3)$ that you describe is essentially the natural extension of $O(2)$ to $O(3)$, namely the subspace of $O(3)$ preserving the $z$-coordinate, except where one "forgets" the factor of $-1$ in the decomposition $-\omega$ ($\omega \in SO(3)$) of the improper rotations. These two groups are isomorphic. One thing I still have not sorted out, which is not so relevant for my purpose, is whether the list of irreps of $G$ described in this post is exhaustive. I do not know how one would establish that it is, or, if it is not, find the others.

Excluding extremely weak parity-violation effects associated with the weak nuclear interaction, one can in fact assign definite parities to molecule eigenstates to describe their behavior under inversion of the spatial coordinates. That is of course not to say that every molecule has an inversion symmetric equilibrium structure. Ammonia (NH3) is a good example, which possesses large "inversion splittings" in its infrared spectra, with state pairs differentiated by their parity under inversion, even though its equilibrium structure is not inversion symmetric (though it is achiral).