We investigate modifications of $\bar{D}$ mesons in nuclear matter with a partial restoration of chiral symmetry. Nuclear matter is constructed by parity doublet model and we fully respect chiral symmetry in our calculations. In the parity doublet model, we have a free parameter which is called a chiral invariant mass defined by a part of the nucleon mass invariant with respect to chiral symmetry. Then, we especially study a chiral invariant mass dependence of spectral function for $\bar{D}_0^*$ meson at normal nuclear matter density. As a result, we find that two clear peaks appear in the spectral function for $\bar{D}_0^*$ meson regarded as a $\bar{D}_0^*$ meson resonance and a threshold enhancement. The peak position of the $\bar{D}_0^*$ meson resonance moves to the lower energy regime as we increase the value of the chiral invariant mass while that of the threshold enhancement moves to the higher energy side. These changes provide us fruitful information of the value of the chiral invariant mass as well as the magnitude of partial restoration of chiral symmetry at normal nuclear matter density.