You can not forge authorization through social. network, because during authorization, for example, through the user's VKontakte, redirects to the site specified in the application settings (to be more precise, to one of the sites from the list of "trusted sites"). VKontakte has a rather confusing authorization scheme , and you will not be able to authorize a user on another site without sending it to him.
If you just want to parse information that is protected by authorization, and if you only need one account for this, then try logging in through a browser, and then copy the received cookies. Well, then just do GET requests to the site, feeding your cURL received cookies. If the site developer is not experienced enough, then this scheme will work.
But in general, for the future: what you are doing is not very correct. Making copies of sites with fake authorization is a sin for which you can get a hand, at least from your hoster. Probably for this reason, and you have instructed cons.