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By ultrafast x-ray diffraction we quantify the strain from coherent and incoherent phonons generated by one- and two-photon absorption. We investigated the ferrimagnetic insulator bismuth-doped yttrium iron garnet, which is a workhorse for laser-induced spin dynamics that may be excited indirectly via phonons. We identify the two-photon absorption by the quadratic intensity dependence of the transient strain and confirm a short lifetime of the intermediate state via the inverse proportional dependence on the pump-pulse duration. We determine the two-photon absorption coefficient using the linear relation between strain and absorbed energy density. For large intensities of about 1TW/cm^{2} considerable strain amplitudes of 0.1% are driven exclusively by two-photon absorption.