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Among topological descriptors, connectivity indices are very important and they have a prominent role in chemistry. The atom-bond connectivity index of a connected graph G is defined as ABC(G)= where dv denotes the degree of vertex v of G and the eccentric connectivity index of the molecular graph G is defined as ξ(G)=, where ε(v) is the largest distance between v and any other vertex u of G. Also, the eccentric atom-bond connectivity index of a connected graph G is equal to ABC5(G)=. In this present paper, we compute this new Eccentric and Connectivity index for an infinite family of Linear Polycene Parallelogram Benzenoid.