  We study the effect of the isovector-scalar meson $a_0$(980) on the
properties of nuclear matter and the neutron star (NS) matter by constructing a
parity doublet model with including the $a_0$ meson based on the chiral
SU(2)$_L\times$SU(2)$_R$ symmetry. We also include the $\omega$-$\rho$ mixing
contribution to adjust the slope parameter at the saturation. We find that,
when the chiral invariant mass of nucleon $m_0$ is smaller than about 800 MeV,
the existence of $a_0$(980) enlarges the symmetry energy by strengthening the
repulsive $\rho$ meson coupling. On the other hand, for large $m_0$ where the
Yukawa coupling of $a_0$(980) to nucleon is small, the symmetry energy is
reduced by the effect of $\omega$-$\rho$ mixing. We then construct the equation
of state (EoS) of a neutron star matter to obtain the mass-radius relation of
NS. We find that, in most choices of $m_0$, the existence of $a_0$(980)
stiffens the EoS and makes the radius of NS larger. We then constrain the
chiral invariant mass of nucleon from the observational data of NS, and find
that $580 \,\text{ MeV} \lesssim m_0 \lesssim 860 \,\text{ MeV} $ for
$L_0=57.7$ MeV.
