Welcome to the forum.
You are allowed to install and Activate PrE on two personal computers, provided that they are not run at the same time.
Are all the computers involved Windows or all Mac?
If you do an online purchase, the installation files that you purchase would be expected to be specific for the computer operating system to which the download is made...either Windows OR Mac. So if your download was to Windows computer and you wanted to install those installation files to a Mac, then no go whether or not you are trying to place Premiere Elements on a 2nd or 3rd computer.
If you purchased from a box packaging, then typically you will get an installation disc for Windows and an installation disc for Mac.
So, those details would factor into your what you want to, coupled with the generalization that Hunt presents regarding Adobe allowing the purchased serial number on no more than two of your computer (not run at same time).
If clarification needed on what I wrote, please do not hesitate to ask.