  Based on 10 billion $J/\psi$ events collected at the BESIII experiment, a
search for $CP$ violation in $\Lambda$ decay is performed in the difference
between $CP$-odd decay parameters $\alpha_{-}$ for $\Lambda \rightarrow p\pi^-$
and $\alpha_{+}$ for $\bar\Lambda \rightarrow \bar{p}\pi^+$ by using the
process $e^+e^- \to J/\psi \rightarrow \Lambda \bar\Lambda $. With a
five-dimensional fit to the full angular distributions of the daughter baryon,
the most precise values for the decay parameters are determined to be
$\alpha_{-} = 0.7519 \pm 0.0036 \pm 0.0024$ and $\alpha_{+} = -0.7559 \pm
0.0036 \pm 0.0030$, respectively. The $\Lambda$ and $\bar{\Lambda}$ averaged
value of the decay parameter is extracted to be $\alpha_{\rm{avg}} = 0.7542 \pm
0.0010 \pm 0.0024$ with unprecedented accuracy. The $CP$ asymmetry
$A_{CP}=(\alpha_{-}+\alpha_{+})/(\alpha_{-}-\alpha_{+})$ is determined to be
$-0.0025 \pm 0.0046 \pm 0.0012$, which is one of the most precise measurements
in the baryon sector. The reported results for the decay parameter will play an
important role in the studies of the polarizations and $CP$ violations for the
strange, charmed and beauty baryons.
