According to this question, the "could not parse proxy url" bug was solved more than 2 years ago.
However, when using the client-side OAuth flow, the aforementioned error message appears when executing
SE.authenticate. Moreover, neither of
error functions within
SE.authenticate are executed. Instead, the user will be redirected to a page that tells him:
Application Login Failure
An error occurred while login into an application.
error description: could not parse proxy url
although the login is successful, as the username appears in the page top navbar.
I encountered this error while developing a Firefox add-on that uses Stack Exchange API. You can confirm the bug by visiting this SE documentation page and clicking on the demo "Run Example" button at the bottom of the page.
The bug takes place only when the user is not already logged in. Otherwise, he won't see this error.