The QS parameter in marketing cloud

you can find most elements that the QS parameter encrypts individually, but you cannot find the encrypted form anywhere except in the email itself.

Starting from there: what you can do is log the value of the entire rendered link in a data extension at the time of send next to the email address. With some delay (e.g. hourly), you could send that link and the email address to an external system from the send log data extension. Real time during sendout, probably not as you would run into performance and reliability issues in case the connection breaks.

As a quick test: Set up a DE "logDE" with two fields, email and profilecenter. Add this to a message, send a preview out to yourself.

%%[

SET @profileURL = profile_center_url
SET @upsert = UpsertDE("logde",1,"email",emailaddr,"profilecenter",@profileURL)

]%%

Result: your email and your profile center link are in the data extension. Set up an hourly automation to do whatever you want with it.

Your sendout performance will be slowed technically, but not to an alarming extent, depending on the size of your sendouts, maybe not even noticeably so. Be sure to load test in case minutes matter for your sendouts, but generally speaking, this approach should work also for sendouts in the millions.