= Solution
The operator $b^s(p)$ is a <fermionic annihilation operator> for a particle with momentum $\mathbf p$ and spin label $s$; $d^{s\dagger}(p)$ is a <fermionic creation operator> for the corresponding <antiparticle>. The c-number <Dirac spinors> $u^s(p)$ and $v^s(p)$ solve the positive-energy particle and antiparticle equations respectively. They supply the spinor wavefunctions in the <mode expansion of a Dirac field>; they are not creation or annihilation operators.
Apply the specified <parity> transformation to the two operators in the <mode expansion of a Dirac field>:
$$
\widehat P\psi(x)\widehat P^{-1}=\eta_P\sum_{p,s}\left[b^s(p_P)u^s(p)e^{-ip\cdot x}-d^{s\dagger}(p_P)v^s(p)e^{ip\cdot x}\right].
$$
Relabel $k=p_P$. The momentum sum is unchanged, $p\cdot x=k\cdot x_P$, and the spinor identities give $u^s(p)=\gamma^0u^s(k)$ and $v^s(p)=-\gamma^0v^s(k)$. The two minus signs in the antiparticle term cancel. Therefore
$$
\boxed{\widehat P\psi(x)\widehat P^{-1}=\eta_P\gamma^0\psi(x_P).}
$$
Taking the <Hermitian conjugate> and using $(\gamma^0)^\dagger=\gamma^0$, $(\gamma^0)^2=1$ in the <Dirac adjoint> yields
$$
\boxed{\widehat P\bar\psi(x)\widehat P^{-1}=\eta_P^*\bar\psi(x_P)\gamma^0.}
$$
The unit-modulus intrinsic-parity phase cancels from all neutral <fermion bilinears>.
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