There is no option for a new serial number thru these forums and Adobe Customer Support does not provide support for older products. Your only option to get the number from Adobe is thru Adobe Customer Support.
For the link below click the Still Need Help? option in the blue area at the bottom and choose the chat option...
Make sure you are logged in to the Adobe site, have cookies enabled, clear your cookie cache. If it fails to connect try using a different browser.
Serial number and activation chat support (non-CC)
https://helpx.adobe.com/contact.html?step=PHSP-PHXS_downloading-installing-setting-up_licensing-activation_stillNeedHelp
Find your Adobe product serial number quickly
If you still have the software installed on the older machine you can try getting the serial number thru that.
If it's a Windows machine then try running Belarc Advisor
http://www.belarc.com/free_download.html
For a Mac you can try:
https://mac-product-key-finder.com/