We present high sensitivity 12/13CO(1-0) molecular line maps covering the full extent of the parsec scale L1551 molecular outflow, including the redshifted east-west (EW) flow. We also present 12CO(3-2) data that extends over a good fraction of the area mapped in the 1-0 transition. We compare the molecular data to widefield, narrow-band optical emission in Hα. While there are multiple outflows in the L1551 cloud, the main outflow is oriented at 50\arcdeg position angle and appears to be driven by embedded source(s) in the central IRS 5 region. The 3-2 data indicate that there may be molecular emission associated with the L1551 NE jet, within the redshifted lobe of main outflow. We have also better defined the previously known EW flow and believe we have identified its blueshifted counterpart. We further speculate that the origin of the EW outflow lies near HH 102. We use velocity dependent opacity correction to estimate the mass and the energy of the outflow. The resulting mass spectral indices from our analysis, are systematically lower (less steep) than the power law indices obtained towards other outflows in several recent studies that use a similar opacity correction method. We show that systematic errors and biases in the analysis procedures for deriving mass spectra could result in errors in the determination of the power-law indices. The mass spectral indices, the morphological appearance of the position-velocity plots and integrated intensity emission maps of the molecular data, compared with the optical, suggest that jet-driven bow-shock entrainment is the best explanation for the driving mechanism of outflows in L1551. The kinetic energy of the outflows is found to be comparable to the binding energy of the cloud and sufficient to maintain the turbulence in the L1551 cloud.