Whatever you do, never give out information like a serial number, anywhere... advertising your bar code isn't going to be a problem for you.
If you can log into your account in the Adobe site, you should be able to locate your serial number in your account information for the products you have registered/activated.
If you cannot find it, then you need to contact Adobe Support either by chat or via phone when you have serial number and activation issues.
Here are some links to help make contact:
http://www.adobe.com/support/chat/ivrchat.html
http://www.adobe.com/support/download-install/supportinfo/
If this software came with your printer, whoever offered it with the printer is responsible for having to supply the serial number. Usually it will be on the packaging of the disc. You may need to contact the seller though you might not have much luck there if you bought it from a store where they would not have an account in your name.