Balkanski and Singer [5] recently initiated the study of adaptivity (or parallelism) for constrained submodular function maximization, and studied the setting of a cardinality constraint. Very recent improvements for this problem by Balkanski, Rubinstein, and Singer [6] and Ene and Nguyen [21] resulted in a near-optimal $(1-1/e-ε)$-approximation in $O(\log n/ε^2)$ rounds of adaptivity. Partly motivated by the goal of extending these results to more general constraints, we describe parallel algorithms for approximately maximizing the multilinear relaxation of a monotone submodular function subject to packing constraints. Formally our problem is to maximize $F(x)$ over $x \in [0,1]^{n}$ subject to $Ax \le 1$ where $F$ is the multilinear relaxation of a monotone submodular function. Our algorithm achieves a near-optimal $(1-1/e-ε)$-approximation in $O(\log^2 m \log n/ε^4)$ rounds where $n$ is the cardinality of the ground set and $m$ is the number of packing constraints. For many constraints of interest, the resulting fractional solution can be rounded via known randomized rounding schemes that are oblivious to the specific submodular function. We thus derive randomized algorithms with poly-logarithmic adaptivity for a number of constraints including partition and laminar matroids, matchings, knapsack constraints, and their intersections.