If you were neither redirected to a "Thank you" page with the license key after the purchase, nor have you received an email with it, please reach out.
First, please double-check that the key is correct, especially if you are not copy and pasting it.
If that doesn't help, make sure that your firewall, ISP or government isn't blocking access to one of these domains:
api.parsify.app (if using Parsify Desktop v.1.6.2 or newer)
api.gumroad.com (if using Parsify Desktop v.1.6.1 or older)
Finally, if the above solutions don't work, please reach out.