There are aspects of parity for particles other than photons that are a bit more complicated than envisaged in the OP.
First of all parity emerges, as most fundamental things in modern physics, from a symmetry principle, namely that of spatial reflections (usually in the sense of reflections of the spatial components of the position vector at the origin of cartesian coordinate system). Then you describe the particles by relativistic quantum fields, building up a representation of the proper orthochronous Poincare group augmented by space reflections.
Now if particles ought to have a specific parity in the strict sense the theory should have spatial reflections as a symmetry. For the hadrons, which can be described in the low-energy realm by an effective (chiral) theory of strongly interacting hadrons (baryons and mesons). The strong interaction is P (even CP) invariant (which is a puzzle in itself) and thus as long as only strong interactions are taken into account parity is a well-defined quantum number of all hadrons. There's even an approximate spontaneously broken symmetry between even and odd parity states with the pions as the pseudo-Goldstone bosons of this broken symmetry. Now since on the fundamental level of QCD the spontaneous breaking is due to the vacuum expectation value of the scalar order parameter ##\langle \bar{\psi} \psi \rangle## (with ##\psi## describing up and down quarks an their antiparticles), the quark condensate. The Goldstone modes are necessarily the pseudoscalar modes, i.e., ##\mathrm{SU}(2)_{\text{L}} \times \mathrm{SU}(2)_{\text{L}}## is spontaneously broken to the iso-vector symmetry ##\mathrm{SU}(2)_{\text{V}}##. Thus the pseudo-Goldstone modes, providing an isotriplet of mesons, must be described by pseudo-scalar fields, and all three pions must have the same parity (-).
Now naively there's also an Abelian axial symmetry, ##\mathrm{U}(1)_{\text{A}}##, if you look at the "classical" Lagrangian, but that symmetry is broken by an anomaly, particularly also when taking into account the electromagnetic interaction (which also breaks isospin symmetry by the way). That's why the decay of the neutral pion into two photons is much larger than expected. The decay into 3 photons is highly surpressed if present at all. According to the PDG the partial decay ratio to ##3 \gamma## is ##<3.1 \cdot 10^{-8}##. Within the standard model it doesn't decay to 3 photons, because this would violate C (charge-conjugation) symmetry of the electromagnetic interaction.