When I view source on my ASP.NET page I get the following snippet:
From looking at your page, I don't understand how you "can't think of any reason that [your] page would require post-backs." It's quite clear to me that when clicking on your Parse button you are calling some methods on the server to interpret the passed in value and convert it to a date, then returning that date to the browser so that the user can see it.
Am I right in assuming that you did not originally create this page and have little experience with ASP.Net? It's not that "MS screwed up big time" (as you mentioned in your comment on Jay S's very well done answer that probably deserves a check mark) it's that your requirements seem to have changed and now you need to ensure that the page works when JavaScript is disabled, which is possible to do with ASP.Net, it's just not the natural way for ASP.Net to do things. However, this will likely require a pretty hefty overhaul of the page, as most drastic changes to requirements do.