The (original and renormalized) action $S[\phi,\psi]$ is assumed to respect parity symmetry. In the pseudoscalar Yukawa theory, the pseudoscalar $\phi$ transforms $$P^{-1}\phi({\bf x},t)P ~=~ -\phi(-{\bf x},t) $$ under a parity transformation. Therefore the action $S[\phi,\psi]$ cannot contain a $\phi^3$ term, cf. above comment by Cosmas Zachos. (Here it should be stressed that a Lorentz invariant $\phi^3$ derivative term are also not possible. E.g. a $\phi^3$ derivative term with the 4D Levi-Civita tensor $\epsilon^{\mu\nu\lambda\kappa}$ vanishes by antisymmetry because at least 2 derivatives must land on the same $\phi$, cf. comments by Peter Kravchuk.) The 1PI effective action $\Gamma[\phi_{\rm cl}, \psi_{\rm cl}]$ inherits$^1$ parity-symmetry, so it can also not contain a $\phi_{\rm cl}^3$ term. Hence the 3-point 1PI vertex vanishes $$ \left. \frac{\delta^3\Gamma[\phi_{\rm cl}, \psi_{\rm cl}]}{\delta \phi_{\rm cl}(x)\delta \phi_{\rm cl}(y)\delta \phi_{\rm cl}(z)}\right|_{\phi_{\rm cl}=0=\psi_{\rm cl}}~=~0,$$ cf. OP's question.
References:
- S. Weinberg, Quantum Theory of Fields, Vol. 2, 1996; Section 16.4 p. 77 + Section 17.2 p. 84.
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$^1$ More generally, the 1PI effective action $\Gamma$ inherits affine symmetries of the action $S$ and the path integral measure, cf. e.g. Ref. 1.