Parallel Algorithm for Donor/Recipient Matching - Graph Matching/Optimization
This can be reduced to an instance of the minimum-cost flow problem.

Build a flow network from your graph, with an edge from each donor to each recipient whose maximum flow is $1$ and whose cost is the negation of the value of that matching.  Add a source vertex $s$, with an edge of weight $\text{maxd}(d)$ and cost $0$ to each donor $d \in D$.  Add a sink vertex $t$ with an edge of cost $0$ and maximum weight $\infty$ from each recipient to $t$.  Now compute the minimum-cost flow.  This can be done in polynomial time, so your problem can be solved in polynomial time.

So your question boils down to asking whether there are parallel algorithms for min-cost flow.  Searching the literature will turn up a bunch of algorithms for that.  If you use Google Scholar, you can find a number of papers on that subject.