The internal composition of neutron stars is currently largely unknown. Due to the possibility of phase transitions in quantum chromodynamics, stars could be hybrid and have quark cores. We investigate some imprints of elastic quark phases (only when perturbed) on the dynamical stability of hybrid stars. We show that they increase the dynamical stability window of hybrid stars in the sense that the onset of instabilities happens at larger central densities than the ones for maximum masses. In particular, when the shear modulus of a crystalline quark phase is taken at face value, the relative radius differences between elastic and perfect-fluid hybrid stars with null radial frequencies (onset of instability) would be up to 1%–2%. Roughly, this would imply a maximum relative radius dispersion (on top of the perfect-fluid predictions) of 2%–4% for stars in a given mass range exclusively due to the elasticity of the quark phase. In the more agnostic approach where the estimates for the quark shear modulus only suggest its possible order of magnitude (due to the many approximations taken in its calculation), the relative radius dispersion uniquely due to a quark phase elasticity might be as large as 5%–10%. Finally, we discuss possible implications of the above dispersion of radii for the constraint of the elasticity of a quark phase with electromagnetic missions such as NICER, eXTP, and ATHENA.