What version of Premiere Elements and on what computer operating system? During installation there is typically a need for Adobe ID and Password as well as purchased serial number.
The Adobe Licensing Agreement allows you to install your purchased activated serial number on no more that 2 of your computers. In your case, the purchased activated serial should be the same for the Windows installation as well as the Mac installation. But the Adobe Licensing Agreement goes on to say that they should not be used at the same time.
This is not Adobe. Rather user to user. But, I have seen this myself when reading the Adobe Licensing Agreement that comes with the product.