This feels like an extremely n00b question, but here goes...
I have a series of HTML files with a small amount of HTML content inside each (exported from a live system).
If you can't modify the server's setup, then you're SOL. You have to tell the server that .html files should be treated as PHP scripts. There's no way around this. It won't magically start sending them through the PHP parser unless you tell it to.
On an Apache setup, it's a matter of putting in a configuration directly, such as
AddType application/x-httpd-php .php .htm .html