  In this paper we investigate the parallel texture structures with cofactor
zeros in the charged lepton and neutrino sectors. The textures can not be
obtained from arbitrary leptonic matrices by making weak basis transformations,
which therefore have physical meaning. The 15 parallel textures are grouped as
4 classes where each class has the same physical implications. It is founded
that one of them is not phenomenological viable and another is equivalent to
the texture zero structures extensively explored in previous literature. Thus
we focus on the other two classes of parallel texture structures and study the
their phenomenological implications. The constraints on the physical variables
are obtained for each class, which are essential for the model selection and
can be measured by future experiments. The model realization is illustrated in
a radiated lepton mass model.
