  The standard modules for an affine Lie algebra $\ga$ have natural
subquotients called parafermionic spaces -- the underlying spaces for the
so-called parafermionic conformal field theories associated with $\ga.$
  We study the case $\ga = \widehat{sl}(n+1,\C)$ for any positive integral
level $k \geq 2.$ Generalizing the $\cal Z$-algebra approach of Lepowsky,
Wilson and Primc, we construct a combinatorial basis for the parafermionic
spaces in terms of colored partitions. The parts of these partitions represent
''Fourier coefficients'' of generalized vertex operators (parafermionic
currents) and can be interpreted as statistically interacting quasi-particles
of color $i,\;1\leq i \leq n,$ and charge $s,\; 1\leq s \leq k-1.$ From a
combinatorial point of view, these bases are essentially identical with the
bases for level $k-1$ principal subspaces constructed by the author in [GeI].
In the particular case of the vacuum module, the character (string function)
associated with our basis is the formula of Kuniba, Nakanishi and Suzuki [KNS]
conjectured in a Bethe Ansatz layout.
  New combinatorial characters are established for the whole standard vacuum
$\ga$-modules.
