The acoustic vibrations of gold nanorods coated with palladium were investigated as a function of Pd amount using ultrafast pump-probe spectroscopy. Both the extensional and breathing vibrational modes of the nanorods were coherently excited and detected. This permits precise determination of their periods, which were found to decrease and increase with Pd deposition, for the extensional and vibrational modes, respectively. These opposite behaviors reflect changes of the nanoparticle size and mechanical properties, in agreement with numerical simulations. Comparison of experimental and computed periods yields information on the amount of deposited Pd, providing a novel tool to characterize bicomponent nano-objects for small fractions of one of the components (Pd/Au atomic fraction down to 5\%).