  A systematic analysis on the phenomenologically allowed textures in Hermitian
quark mass matrices with four texture-zeros is presented. The current data
allow a full determination of the mass matrix elements, provided that the
actual number of the parameters in the matrices is less than or equal to ten.
Besides the standard texture with zeros located in (1,1) and (1,3) positions,
we find four new type of parallel mass matrices. All of them are found to have
similar hierarchical structures: three of the nonzero matrix elements are at
the same order of magnitude with a small geometric hierarchy, while the
remaining one is relatively much smaller. These textures show a possibility
that a realistic quark mass matrix may have an approximate flavor
$S(2)_{L}\otimes S(2)_{R}$ symmetry, i.e. a partial flavor democracy. The
absolute values of the CP violating phases in the matrix elements are found to
be either close to their maximum ($\approx 90^\circ$) or very small ($\leq
10^{\circ}$). We also find twenty-six quasi-parallel and seventeen non-parallel
textures allowed by the experimental data.
