The MediaWiki API doesn't understand infoboxes. So, you have basically two options:
Parse the infobox yourself. You can either parse the wikitext directly or the generated HTML table (both are available from the API).
Let somebody else do the parsing. This is exactly what DBPedia does. Wikidata tries to do something similar, but it probably won't contain enough data to be usable for a long time; see growth statistics.