quantum field theory - Irrelevance of parastatistics for space dimension > 2

Consider a system of $n$ undistinguishable particles moving in $d$-dimensional Euclidean space $E^d$. The configuration space is $M=((E^d)^n \setminus \Delta)/S_n$ where $\Delta$ is the diagonal (subspace where at least 2 particles have coincidental positions) and $S_n$ is the group permuting the particles

Quantization of this system yields superselection sectors corresponding to unitary irreducible representations of $\pi_1(M)$: $S_n$ for $d > 2$, $B_n$ for $d = 2$. The trivial representation yields bosonic statistics, the sign representations yield fermionic statistics. For $d > 2$ there are no other 1-dimensional representations. For $d = 2$ there are other 1-dimensional representations in which switching two particles generates an arbitrary phase. These yield anyonic statistics.

What about higher dimensional irreducible representations? These correspond to parastatistics. It is said that for $d > 2$ we can safely ignore them because in some sense they are equivalent to ordinary bosons/fermions. However for $d = 2$ this is not the case. Why?

Why is parastatistics redundant for $d > 2$ but not for $d = 2$?