It is well known that parallels of given smooth hypersurfaces in Euclidean space have Legendrian singularities. If we are concerned with the way in which these parallels change as we alter the distance, perestroikas of wave front sets will appear. This problem has been considered by several people. Roughly speaking, it is shown that generic perestroikas of the singularities of parallels of smooth hypersurfaces are generic perestroikas of Legendrian singularities. V. M. Zakalyukin [J. Sov. Math. 27, 2713-2735 (1984); translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat. 22, 56-93 (1983; Zbl 0534.58014)] proved that generic perestroikas of Legendrian singularities are stable perestroikas of Legendrian singularities in the case \(n\leq 5\) and he classified these perestroikas. In this paper we consider the realization problem of the perestroikas of Legendrian singularities. However, his notion of extended Legendrian submanifolds is too wide to supply this realization problem. Some of the lists in his classification cannot be realized as parallels of hypersurfaces. We give a candidate of this class which we call graphlike Legendrian unfoldings. Roughly speaking, a graphlike Legendrian unfolding is a Legendrian submanifold germ with submersive generating function. The general properties of graphlike Legendrian unfoldings are studied and a realization theorem is proved. All the arguments are considered in the framework of Hamiltonian formalism which also contains the situation of parallels of hypersurfaces in a Riemannian manifold.