The resolution of climate models is limited by computational cost. Therefore, we must rely on parameterizations to represent processes occurring below the scale resolved by the models. Here, we focus on parameterizations of ocean mesoscale eddies and employ machine learning (ML), namely relevance vector machines (RVM) and convolutional neural networks (CNN), to derive computationally efficient parameterizations from data, which are interpretable and/or encapsulate physics. In particular, we demonstrate the usefulness of the RVM algorithm to reveal closed-form equations for eddy parameterizations with embedded conservation laws. When implemented in an idealized ocean model, all parameterizations improve the statistics of the coarse-resolution simulation. The CNN is more stable than the RVM such that its skill in reproducing the high-resolution simulation is higher than the other schemes; however, the RVM scheme is interpretable. This work shows the potential for new physics-constrained interpretable ML turbulence parameterizations for use in ocean climate models.