  In this paper, we study the number of limit cycles that can bifurcate from a
periodic annulus in discontinuous planar piecewise linear Hamiltonian
differential system with three zones separated by two parallel straight lines,
such that the linear differential systems that define the piecewise one have a
center and two saddles. That is, the linear differential system in the region
between the two parallel lines (i.e. the central subsystem) has a center and
the others subsystems have saddles. We prove that if the central subsystem has
a real or a boundary center, then we have at least six limit cycles bifurcating
from the periodic annulus by linear perturbations, four passing through the
three zones and two passing through the two zones. Now, if the central
subsystem has a virtual center, then we have at least four limit cycles
bifurcating from the periodic annulus by linear perturbations, three passing
through the three zones and one passing through the two zones. For this, we
obtain a normal form for these piecewise Hamiltonian systems and study the
number of zeros of its Melnikov functions defined in two and three zones
