The derivatives of option prices with respect to underlying parameters are commonly referred to as Greeks, and they measure the sensitivities of option prices to these parameters. When the closed-form solutions for option prices do not exist and the discounted payoff functions of the options are not sufficiently smooth, estimating Greeks is computationally challenging and could be a burdensome task for high-dimensional problems in particular. The aim of this paper is to develop a new method for estimating option Greeks by using random parameters and leastsquares regression. Our approach has several attractive features. First, just like the finite-difference method, it is easy to implement and does not require explicit knowledge of the probability density function and the pathwise derivative of the underlying stochastic model. Second, it can be applied to options with discontinuous discounted payoffs as well as options with continuous discounted payoffs. Third, and most importantly, we can estimate multiple derivatives simultaneously. The performance of our approach is illustrated for a variety of examples with up to fifty Greeks estimated simultaneously. The algorithm is able to produce computationally efficient results with good accuracy.