In this work, Left-Right Symmetric Model (LRSM) has been realized with the modular group of level 3, that is, \(\Gamma (3)\) and weight 2, which is isomorphic to non-abelian discrete symmetry group \(A_4\) . Here, we take into consideration the possibilities of texture zeros in the mass matrices of the modular symmetric left-right model. As the type-I and type-II seesaw masses appear naturally in LRSM, the texture zero study has been carried out for both the Dirac and heavy right-handed Majorana mass matrices, respectively. The advantage of using modular symmetry is that we do not require the use of any extra particles (flavons) for obtaining the desired results within the realization of the model. As LRSM provides an environment for the study and analysis of beyond Standard Model phenomenology, in this work, we take into consideration the study of contributions of Neutrinoless Double Beta Decay within the framework of minimal LRSM with \(A_4\) modular symmetry.