We consider the elastic scattering problem by multiple disjoint arcs or cracks in two spatial dimensions. A key aspect of our approach lies in the parametric description of each arc's shape, which is controlled by a potentially high-dimensional, possibly countably infinite, set of parameters. We are interested in the efficient approximation of the parameter-to-solution map employing model order reduction techniques, specifically the reduced basis method. Firstly, we use boundary potentials to transform the boundary value problem, originally posed in an unbounded domain, into a system of boundary integral equations set on the parametrically defined open arcs. We adopt the two-phase paradigm (offline and online) of the reduced basis method to construct a fast surrogate. In the offline phase, we construct a reduced order basis tailored to the single arc problem assuming a complete decoupling among arcs. In the online phase, when computing solutions for the multiple arc problem with a new parametric input, we use the aforementioned basis for each individual arc. We present a comprehensive theoretical analysis of the method, fundamentally based on our previous work [Pinto et al., J. Fourier Anal. Appl. 30 (2024) 14]. In particular, the results stated therein allow us to find appropriate bounds for the so-called Kolmogorov width. Finally, we present a series of numerical experiments demonstrating the advantages of our proposed method in terms of both accuracy and computational efficiency.