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 -> p pi(-) and alpha(+) for (Lambda) over bar -> (p) over bar pi(+) by using the process e(+)e(-) -> J/psi -> Lambda(Lambda) over bar. With a five-dimensional fit to the full angular distributions of the daughter baryon, the most precise values for the decay parameters arc determined to be a_ = 0.7519 +/- 0.0036 +/- 0.0024 and alpha(+) = -0.7559 +/- 0.0036 +/- 0.0030, respectively. The Lambda and (Lambda) over bar averaged value of the decay parameter is extracted to be a(avg) = 0.7542 +/- 0.0010 +/- 0.0024 with unprecedented accuracy. The CP asymmetry A(CP) = (alpha_ + alpha(+))/(a_-alpha(+)) is determined to be -0.0025 +/- 0.0046 +/- 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.