Many members in the complex oxide material family are available as small single crystals or epitaxial thin-films only. Therefore, measuring the elastic properties such as the Young's modulus of these materials is a challenging task and little is known about these parameters for many complex oxides. With LaAlO3 and (La,Sr)MnO3 mechanical resonator samples kindly provided by CNR-SPIN, we demonstrate how tensile stress and Young's modulus of a thin-film sample can be determined by carefully characterizing all detectable flexural modes of nanomechanical string resonators fabricated from the film. Applying this method in a cryogenic environment is straightforward, therefore we were able to obtain these material parameters as a function of temperature from 4K to 300K with resolution high enough to resolve the structural phase transition of the SrTiO3 substrate at 105K.