  We construct a coupled wire model for a sequence of non-Abelian quantum Hall
states occurring at filling factors $\nu=2/(2M+q)$ with integers $M$ and
even(odd) integers $q$ for fermionic(bosonic) states. They are termed $Z_2
\times Z_2$ orbifold states, which have a topological order with a neutral
sector described by the $c=1$ orbifold conformal field theory (CFT) at radius
$R_{\rm orbifold}=\sqrt{p/2}$ with even integers $p$. When $p=2$, the state can
be viewed as two decoupled layers of Moore-Read (MR) state, whose neutral
sector is described by the Ising $\times$ Ising CFT and contains a $Z_2 \times
Z_2$ fusion subalgebra. We demonstrate that orbifold states with $p>2$, also
containing a $Z_2 \times Z_2$ fusion algebra, can be obtained by coupling
together an array of MR $\times$ MR wires through local interactions. The
corresponding charge spectrum of quasiparticles is also examined. The orbifold
states constructed here are complementary to the $Z_4$ orbifold states, whose
neutral edge theory is described by orbifold CFT with odd integer $p$ and
contains a $Z_4$ fusion algebra.
