Parallel shear flows cannot be kinematic dynamos on their own (Zel’dovich, Sov. Phys. JETP, vol. 4, 1957, pp. 460–462), but the addition of small flow perturbations can trigger dynamo action. Using an optimization algorithm inspired by Willis (Phys. Rev. Lett., vol. 109 (25), 2012, 251101) and Chen et al. (J. Fluid Mech., vol. 783, 2015, pp. 23–45), we identify the smallest perturbation that when added to Kolmogorov flow can trigger dynamo action at some fixed value of the magnetic Reynolds number. In this way we numerically measure the fragility of the Zel’dovich anti-dynamo theorem. The minimal perturbations have surprisingly simple spatial structures. Their magnitudes vary inversely proportional to the magnetic Reynolds number and are always much larger than theoretical lower bounds calculated here using the methods of Proctor (Geophys. Astrophys. Fluid Dyn., vol. 98 (3), 2004, pp. 235–240; J. Fluid Mech., vol. 697, 2012, pp. 504–510).