I'm wondering what the easiest way to extract only the information contained withing a certain template would be using the wikimedia api.
I'd like to extract the information contained in the template "Template:Mycomorphbox" for this page: http://en.wikipedia.org/wiki/Amanita_phalloides
I'm a bit frustrated that it seems like I have to pull the entire content of the page to get the information that I need. Surely there has to be a better way.
Indeed there is a better way. You must not extract information from templates (or from wikitext in general). That's not your job nor your application's, it's MediaWiki's.
Use Wikidata, which is where the structured information from and for Wikipedia is stored. See the Wikibase API documentation and see some of the properties used for biology stuff or ask if something is unclear.