How do you make a .php contact form work on the 'old' web server?

[#1]

Hello. I am set up for the Plesk server because its the only way I can get my .php contact forms to work (I just drop all the web files into the httpdocs folder and the forms work). This is no the case with the 'old' web server - where instead if you click on the contact from navigational link on the website, it would download the .php file to my computer. I have recommended PlusNet to 7 or 8 people, some of who have websites hosted on the 'old' web server but who want to have contact forms working as well. But now, to upgrade to Plesk they have to pay a minimum of £4.59 extra per month on top of their existing PlusNet subscription. They don't want to pay this when they already have perfectly adequate web space with their standard account (and without data transfer limits).
So my question is, does anyone know how to get the .php contact forms to work on the 'old' webserver when using a web design program such as RapidWeaver, which has a .php contact form 'template' built into it? An example can be seen working here: http://www.typarke.co.uk/contact-us/contact-us.php.
As you can tell, I am not technical...
Thanks very much.

[#2]

Which pls was the 'old' web server? The standard 'home pages' server/s which until very recently all users had space on (free) by default, or the the CCGI servers which you needed to 'turn on' from the account's control panel on the portal before you could start to user one of them?
Both types of server would run a non-active (e.g. non PHP; non database access) website, but only the CCGI servers would run an active one.
It is physically possible for PHP code to be downloaded to the client PC and executed each time you wanted to fill in a form, but this would then entail installing a PHP interpreter and probably a webserver environment on each client PC. For the average user visiting the website, this is simply not going to obtain.

[#3]

I had the CGI server activated but could never make sense of what I had to do. Perhaps I had to put the .php files in there and link to them or something - but I could never make it work. Thanks anyway.