We consider the problem of maximizing the multilinear extension of a submodular function subject a single matroid constraint or multiple packing constraints with a small number of adaptive rounds of evaluation queries. We obtain the first algorithms with low adaptivity for submodular maximization with a matroid constraint. Our algorithms achieve a $1-1/e-ε$ approximation for monotone functions and a $1/e-ε$ approximation for non-monotone functions, which nearly matches the best guarantees known in the fully adaptive setting. The number of rounds of adaptivity is $O(\log^2{n}/ε^3)$, which is an exponential speedup over the existing algorithms. We obtain the first parallel algorithm for non-monotone submodular maximization subject to packing constraints. Our algorithm achieves a $1/e-ε$ approximation using $O(\log(n/ε) \log(1/ε) \log(n+m)/ ε^2)$ parallel rounds, which is again an exponential speedup in parallel time over the existing algorithms. For monotone functions, we obtain a $1-1/e-ε$ approximation in $O(\log(n/ε)\log(m)/ε^2)$ parallel rounds. The number of parallel rounds of our algorithm matches that of the state of the art algorithm for solving packing LPs with a linear objective. Our results apply more generally to the problem of maximizing a diminishing returns submodular (DR-submodular) function.