# Ambiguity in assigning intrinsic parity

We know that, fermions can have intrinsic parity either $\eta_P=+1$ or $=-1$. How does one then fix the intrinsic parities ofthe elementary particles, uniquely? Again, the intrinsic parity of a baryon is the product of the intrinsic parities of the quarks. But unless one can fix the intrinsic parity of the quarks, uniquely, it remains ambiguous. Should we take $\eta_P=+1$ for all of them or $\eta_P=-1$ for all of them? How to fix this?

If you're referring to elementary fermions (described by the Dirac equation), there is no ambiguity: fermions have an intrinsic parity +1 while antifermions have -1. This is easy to establish by starting from the Dirac equation, you deduce that the parity operator is $\gamma^0$ and using the general solutions for spinors $u$ (particles) and $v$ (anti-particles) you can check that:
$$u(-\vec{p}) = \gamma^0 u(\vec{p}), ~~~v(-\vec{p}) = -\gamma^0 v(\vec{p})$$
hence, the sign of the intrinsic parity. For the parity of the baryons, it's not so straightforward because it is defined as the product of the intrinsic parity of the 3 quarks (giving +1) and the product of the angular momentum between 2 of the 3 quarks (giving $(-1)^{l_{1,2}}$) and the angular momentum between the third and the barycenter of the first 2 (giving another $(-1)^{l_{3,1-2}}$).