  In this paper we investigate the parallel texture structures containing
texture zeros in charged lepton mass matrix $M_{l}$ and cofactor zeros in
neutrino mass matrix $M_{\nu}$. These textures are interesting since they are
related to the $Z_{n}$ flavor symmetries. Using the weak basis permutation
transformation, the 15 parallel textures are grouped as 4 classes (class
I,II,III and IV) with the matrices in each class sharing the same physical
implications. Under the current experimental data, the class I, III with
inverted mass hierarchy and class II with normal mass hierarchy are
phenomenologically acceptable. The correlations between some important physical
variables are presented, which are essential for the model selection and can be
text by future experiments. The model realization is illustrated by means of
$Z_{4}\times Z_{2}$ flavor symmetry.
