In this work, it is considered that the vector space is composed by the infinitesimal deformations of the model ${\mathbb{Z}_3}$-filiform Lie algebra ${L^{n,m,p}}$. By using these deformations, all the ${\mathbb{Z}_3}$-filiform Lie algebras can be obtained, hence the importance of these deformations. The results obtained in this work, together with those obtained by Khakimdjanov and Navarro ( J. Geom. Phys. 2011 and 2012), lead to compute the total dimension of the mentioned space of deformations.