I wrote the Wikipedia page in question, so I feel bad. I thought it was clear.
There is a recent textbook by Banks which covers the spin/statistics theorem pretty good. I hope it is ok. The main difficulty is that there is no quantum field theory book that covers analytic continuation to Euclidean space, and this is the essential thing.
This is worked out by each person on their own, as far as I know. The problem is that it is very easy to say "plug in i times t everywhere you see t" and get 90% of everything right, without understanding anything. Streater and Whitman do it, that's most of their book, but they are too formal to be comprehensible. Schwinger is too long ago (and ideosyncratic). Perhaps the statistical section of Feynman and Hibbs (Path integrals), where they actually rederive the path integral in imaginary time, will allow you to extrapolate to the general bosonic fields.
The Fermionic case requires the Euclidean continuation of Majorana spinors, and this was in the literature more recently: http://arxiv.org/abs/hep-th/9608174. This stuff is covered in none of the textbooks, and unfortunately I can't recommend any of them with a good conscience.
Later Edit: If you don't want to go to Euclidean space, you should avoid anything past Feynman/Schwinger. The best path then is possibly to work through Pauli's argument:W. Pauli, The Connection Between Spin and Statistics, Phys. Rev. 58, 716-
722(1940).
The (orbital) wave function for $l=1$ doesn't just "have to be" antisymmetric. It demonstrably "is" antisymmetric. The relevant part of this wave function is completely determined, it's a particular function, so we may see whether it's symmetric or antisymmetric and indeed, it's the latter.
Two particles – in this case two pions – orbiting each other are described by a wave function of the relative position
$$\vec r = \vec r_1 - \vec r_2 $$
Writing $\vec r$ in spherical coordinates, a well-defined $l,m$ means that the wave function factorizes to
$$ \psi (\vec r) = \psi_r(r)\cdot Y_{lm}(\theta,\phi)$$
The angular dependence simply has to be given by $Y_{lm}$ because $Y_{lm}$ is, up to the overall normalization, the only wave function with the right angular momentum and its $z$-component being $l,m$.
But $Y_{lm}$ is easily seen to be an odd function of $\hat r$ for odd $l$ and even function of $\hat r$ for even $l$. In particular, $Y_{00}$ is a constant while $Y_{10}$ and $Y_{1,\pm 1}$ are proportional to $z$ and $x\pm iy $ on the unit sphere, respectively.
These functions proportional to $x,y,z$ are clearly odd functions of $\hat r$, so they change the sign under $\vec r\to -\vec r$ which is the sign flip equivalent to $\vec r_1\leftrightarrow \vec r_2$. This odd parity is also called antisymmetry.
Best Answer
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: \begin{equation} \psi(x_1,x_2) = - \psi(-x_1,-x_2). \end{equation}
Being odd under interchange of particles says \begin{equation} \psi(x_1, x_2) = - \psi(x_2, x_1). \end{equation}
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).