I do not use ASP.Net myself so I can't give specific help on that.
I prefer multi-platform and open source solutions over Microsoft tools for my own developments.
For web development my preferred tools are Apache web server and PHP.
If you can run external executable programs from your ASP.Net web page (there is most probably way to do that) then you should be able rewrite he control software using ASP.Net technologies if you know the ASP.Net programming.
The C# programs can't directly access parallel port. If you want to access parallel port from C# program you need to use some external library or control program to do the actual controlling.
If you get something nice working example you want to share to the world, I would be happy to publish also that example on m web pages.