Why does the proton have a parity $P=1$ and an anti-symmetric wave function? The overall parity of a proton is 1 because the parity of a quark is 1. How does this go together with the proton's wave function being anti-symmetric?
Is the reason for the proton's wave function's anti-symmetry the fact that in $SU(3)_C$ you consider the $u,d,s$ quark flavors to be identical for the strong interaction?
 A: The spin-flavor part of the proton wavefunction,
$$
|p_\uparrow\rangle= \frac{1}{\sqrt {18}} [ 2| u_\uparrow d_\downarrow u_\uparrow \rangle + 2| u_\uparrow  u_\uparrow d_\downarrow \rangle +2| d_\downarrow u_\uparrow   u_\uparrow \rangle  \\ - | u_\uparrow u_\downarrow d_\uparrow\rangle  -| u_\uparrow d_\uparrow u_\downarrow\rangle  -| u_\downarrow d_\uparrow u_\uparrow\rangle  
-| d_\uparrow u_\downarrow u_\uparrow\rangle  -| d_\uparrow  u_\uparrow u_\downarrow\rangle   -| u_\downarrow u_\uparrow d_\uparrow\rangle   ]
$$
is completely symmetric under interchange of any two quarks with each other; even though, individually, flavor and spin interchanges are of mixed symmetry: this is the cornerstone of the model, really.
The position (orbital angular momentum) wavefunction is then an S-wave, symmetric, so, given the total antisymmetrization by the color, normally omitted, the three fermion constituent quarks are in a fully antisymmetric state, as they should be.
Consequently, since S contributes + to the overall parity,  this parity then turns out to be ++++ $\leadsto$ + for the overall parity of the proton.
The idea linking generalized Pauli antisymmetrization of fermions to parity amounts to appreciating that there is no P wave spatial component, so the parity of the nucleon is the product of the parities of the three constituents.
A: The spatial parity of the proton (and the quark) is +1 by convention. Fermion and antifermion have opposite spatial parity. So the antiproton has spatial parity -1, by the same convention.   But it could equally well be the other way about, provided you're consistent.
The proton wave function is antisymmetric under permutation of labels, as @CosmasZachos says.  So is the antiproton wave function.  That's not a convention, that's the spin statistics theorem.
There is no direct link (or conflict) between spatial parity and permutation symmetry here.
