Here is the API call that would get you structure unionization data by region.
http://api.brain-map.org/api/v2/data/query.json?criteria=model::StructureUnionize,rma::criteria,structure[acronym$eq'DR']
This returns an entry for each of the tens of thousands of section data sets (experiments) in the atlas, so it is likely that you will want to refine it further to get just what you need. There is lots of documentation available such as the links in Jeremy’s post above and these:
http://help.brain-map.org/display/api/Allen%2BBrain%2BAtlas%2BAPI
http://help.brain-map.org/display/api/RESTful+Model+Access+(RMA)