= Solution
Writing $x_P=(t,-\mathbf x)$, the standard <parity transformation of a Dirac field> is
$$
\boxed{\psi'(x)=\eta_P\gamma^0\psi(x_P),\qquad \bar\psi'(x)=\eta_P^*\bar\psi(x_P)\gamma^0,\qquad |\eta_P|=1.}
$$
Since $\gamma^0\gamma^5\gamma^0=-\gamma^5$, the <fermion bilinear> $\bar\psi\gamma^5\psi$ changes sign. Invariance of the <pseudoscalar Yukawa interaction> therefore requires
$$
\boxed{\phi'(x)=-\phi(x_P).}
$$
Thus $\phi$ is a <pseudoscalar>, despite having the usual scalar-form <kinetic term> and <mass term>. In this convention $(\gamma^5)^\dagger=-\gamma^5$, but $\bar\psi\gamma^5\psi$ is Hermitian because of the additional $\gamma^0$ in the <Dirac adjoint>. A real $g$ is consequently consistent with a Hermitian <Lagrangian density>.
In <perturbation theory> about the parity-invariant <vacuum state>, <renormalization> can preserve this non-anomalous <parity symmetry in quantum field theory>. Every local <counterterm> must then be parity even, at every <loop order>. The allowed terms are precisely $\mathcal L_{\rm ct}^{(+)}$ in <1/b>; all coefficients in $\mathcal L_{\rm ct}^{(-)}$ are zero. In particular, \b[no $\phi$ or $\phi^3$ counterterm is generated], whereas $\phi^4$ is allowed and required. The one-point and three-point pure-boson vertices also vanish to all orders: a parity-odd <Lorentz scalar> cannot be formed from their at most two independent external <four-momenta>. This last reasoning is specific to these low-point vertices; it does not assert that <parity symmetry in quantum field theory> is an independent $\phi\mapsto-\phi$ internal symmetry with the fermions held fixed.
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