Parity Transformations on Molecular Orbitals

I’ve been looking into the quantum mechanics of MO Theory and I stumbled across parity symmetries of atomic and molecular wavefunctions.

In short, I understand how the parity operator $hat{mathcal{P}}$ acts on single hydrogen-like wavefunctions of atomic orbitals (specifically only the angular part) by a mapping of polar coordinates $hat{mathcal{P}}(r, theta, phi) mapsto (r, pi-theta, pi+phi)$, resulting in:

$$hat{mathcal{P}}Y_{ell}^m(theta, phi) = (-1)^{ell}Y_{ell}^{m}(theta, phi)$$

This would denote s-orbitals to be of even parity and p-orbitals to be odd. However I encounter a problem when I try to repeat the process with molecular orbitals i.e.: through the LCAO method. Taking H$_2$ as an example, with two 1s orbitals interacting with each other, the bonding/antibonding MOs of H$_2$ can be written as:

$$Psi_{sigma} = psi_{1s}(r_1, theta_1, phi_1) + psi_{1s}(r_2, theta_2, phi_2)
Psi_{sigma*} = psi_{1s}(r_1, theta_1, phi_1) – psi_{1s}(r_2, theta_2, phi_2)
$$

When applying $hat{mathcal{P}}$ to both of them, I get even (or gerade) symmetry for both MOs, which are because both angular parts of the 1s orbitals are even regardless of in-phase/out-of-phase combinations. But it is well-established that the antibonding $sigma^*$ MO is of ungerade symmetry! This also similarly happens with $sigma$/$sigma^*$ bonding MOs of p$_z$ orbitals, where both parity operations result in ungerade parity even though the bonding MO should be gerade.

The question(s) is: Where am I going wrong here? Am I using $hat{mathcal{P}}$ incorrectly? Is it appropriate to use it for MOs if it works on AOs? I am aware that there is a separate ‘inversion operator’ $hat{mathbb{i}}$ for centrosymmetric molecules that determines g/u symmetry but I have yet to find any mathematical formulation of this operator (if one exists). And if it does exist, then how does it fit in with LCAO MO wavefunctions?