Fermion wavefunctions are antisymmetric under the interchange of two particles. Spatial inversion flips the spatial coordinate, but does not interchange particles.
In other words, let's say we have a two particle wave function, $\psi(x_1, x_2)$ (where $x_1$ is the position of particle 1, and $x_2$ is the position of particle 2).
Being odd under parity says:
\psi(x_1,x_2) = - \psi(-x_1,-x_2).
Being odd under interchange of particles says
\psi(x_1, x_2) = - \psi(x_2, x_1).
Thus parity and statistics are independent properties. In particular, it is perfectly consistent to have a parity odd boson.
(Things get a little more interesting if you have spin, because parity also affects the polarization, but that seems like a more complicated question than what you asked).