In the minimum cost submodular cover problem (MinSMC) problem, given a monotone nondecreasing submodular function f 2^V →ℤ^+, a cost function c: V→ℝ^+, and an integer k≤ f(V), the goal is to find a subset A⊆ V with the minimum cost such that f(A)≥ k. The MinSMC can be found at the heart of many machine learning and data mining applications. In this paper, we design a parallel algorithm for MinSMC that obtains a solution with an approximation ratio of at most H(min{Δ,k})/1-5ε with a probability of 1-3ε in O(log mlog nlog^2 mn/ε^4) rounds, where Δ=max_v∈ Vf(v), H(·) is the Harmonic number, n=f(V), m=|V|, and ε is a constant in (0,1/5). This paper is the first to obtain a parallel algorithm for the weighted version of the MinSMC problem with an approximation ratio arbitrarily close to H(min{Δ,k}).