  We consider parallel, or low adaptivity, algorithms for submodular function
maximization. This line of work was recently initiated by Balkanski and Singer
and has already led to several interesting results on the cardinality
constraint and explicit packing constraints. An important open problem is the
classical setting of matroid constraint, which has been instrumental for
developments in submodular function maximization. In this paper we develop a
general strategy to parallelize the well-studied greedy algorithm and use it to
obtain a randomized $\left(\frac{1}{2} - \epsilon\right)$-approximation in
$\operatorname{O}\left( \frac{\log^2 n}{\epsilon^2} \right)$ rounds of
adaptivity. We rely on this algorithm, and an elegant amplification approach
due to Badanidiyuru and Vondr\'ak to obtain a fractional solution that yields a
near-optimal randomized $\left( 1 - 1/e - \epsilon \right)$-approximation in
$O\left( {\frac{\log^2 n}{\epsilon^3}} \right) $ rounds of adaptivity. For
non-negative functions we obtain a $\left( {3-2\sqrt{2}}\right)$-approximation
and a fractional solution that yields a $\left( {\frac{1}{e} -
\epsilon}\right)$-approximation. Our approach for parallelizing greedy yields
approximations for intersections of matroids and matchoids, and the
approximation ratios are comparable to those known for sequential greedy.
