We investigate coherent phonon oscillations of BaFe$_2$As$_2$ using optical pump-probe spectroscopy. Time-resolved optical reflectivity shows periodic modulations due to $A_{1g}$ coherent phonon of $c$-axis arsenic vibrations. Optical probe beams polarized along the orthorhombic $a$- and $b$-axes reveal that the initial phase of coherent oscillations shows a systematic deviation as a function of temperature, although these oscillations arise from the same $c$-axis arsenic vibrations. The oscillation-phase remains anisotropic even in the tetragonal structure, reflecting a nematic response of BaFe$_2$As$_2$. Our study suggests that investigation on the phase of coherent phonon oscillations in optical reflectivity can offer unique evidence of a nematic order strongly coupled to a lattice instability.
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