Your browser parses, yes, but then it goes and fetches images, css, javascript, iframes, etc. Following the analogy, these are 'api calls'. It knows to make these calls because the HTML defined the links, and for the HTML to be rendered the contents of the links are required.
I maintain the browser is the client - it responds to your actions, but it does so with a lot more activity than just parsing.
>What makes hypermedia interesting IMHO is (minor point) easier intra-linking within a webservice, and (major point) the ability within a service to point to resources from another one. > >This will enable the distributed web we are all waiting for.
Couldn't agree more. But without some standards, those links to other services may not be as useful as they could be.