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We observe pronounced oscillations in differential reflectivity of 9 nm and 60 nm BaFe\textsubscript{2}As\textsubscript{2} (Ba-122) thin films using ultrafast optical spectroscopy. Our studies show that the oscillations result from propagating longitudinal acoustic (LA) phonon wave packets with strong thickness and temperature dependence. Particularly, the experimentally measured oscillation frequency approaches to 50 GHz for the ultra-thin film. Our calculations show that Young's modulus of 9 nm thin film is nearly four times as large as that of 60 nm thin film, consistent with the experiment. The increase in Young's modulus as thickness decrease was attributed to the decrease in parent Ba-122 tetragonality $c/a$ near the film-substrate interface due to material-substrate mismatch effect. %Temperature dependence of LA phonon mode frequency for 9 nm Ba-122 thin film is reported. The temperature-dependent change in LA phonon frequency was attributed to the change in parent Ba-122 othorhombicity $(a-b)/(a+b)$.