This is the first of a series of three papers about the Elastic Manifold model. This classical model proposes a rich picture due to the competition between the inherent disorder and the smoothing effect of elasticity. In this paper, we prove a Parisi formula, i.e. we compute the asymptotic quenched free energy and show it is given by the solution to a certain variational problem. This work comes after a long and distinguished line of work in the Physics literature, going back to the 1980's (including the foundational work by Daniel Fisher [29], Marc Mezard and Giorgio Parisi [50, 51], and more recently by Yan Fyodorov and Pierre Le Doussal [34, 35]. Even though the mathematical study of Spin Glasses has seen deep progress in the recent years, after the celebrated work by Michel Talagrand [67, 68], the Elastic Manifold model has been studied from a mathematical perspective, only recently and at zero temperature. The annealed topological complexity has been computed, by the first author with Paul Bourgade and Benjamin McKenna [15, 16]. Here we begin the study of this model at positive temperature by computing the quenched free energy. We obtain our Parisi formula by first applying Laplace's method to reduce the question to a related new family of spherical Spin Glass models with an elastic interaction. The upper bound is then obtained through an interpolation argument initially developed by Francisco Guerra [42] for the study of Spin Glasses. The lower bound follows by adapting the cavity method along the lines explored by Wei-Kuo Chen [23] and the multi-species synchronization method of Dmitry Panchenko [55]. In our next papers [19, 20] we will analyze the consequences of this Parisi formula.