Do you already have PM installed and in current use on two computers? If so, then yes I believe you will need to purchase a new licence for the third activation. Email infoservice@magix.net and ask whether you qualify for any sort of "special price".
OTOH, if one of those activations is no longer needed, go to Help>Deactivate Program on the computer where it is no longer needed. Allow a few minutes for the Magix servers to "catch up" and then attempt to activate the program on the new installation.