Two parallel Gaussian vortices of circulations Γ1 and Γ2 radii a1 and a2, separated by a distance b may become unstable by the elliptical instability due the elliptic deformation of their cores. The goal of the paper is to analyse this occurrence theoretically in a general framework. An explicit formula for the temporal growth rate of the elliptical instability in each vortex is obtained as a function of the above global parameters of the system, the Reynolds number Γ1/v and the non-dimensionalized axial wavenumber kzb of the perturbation. This formula is based on a known asymptotic expression for the local instability growth rate at an elliptical stagnation point which depends on the local characteristics of the elliptical flow and the inclination angle of the local perturbation wavevector at this point. The elliptical flow characteristics are estimated by considering each Gaussian vortex alone in a weak uniform external strain field whose properties are provided by a point vortex modelling of the vortex pair. The inclination angle is obtained from the dispersion relation for the Gaussian vortex normal modes and the local expression near each vortex centre for the two helical modes of azimuthal wavenumber m = 1 and m = −1 which constitute the elliptical instability global mode. Both the final formula and the hypotheses made for its derivation are tested and validated by direct numerical simulations and large-eddy simulations.