Confirmed.

Most likely the extension's parser instance isn't passing links
back up to the parent parser.

extension/Cite/Cite.php had been using its own Parser instance due
to some prior bugs with extension handling and the unique index.
(Avar is unable provide details of this.)

These bugs appear to no longer be present, as far as I can tell.
But who knows, with no bug report and no test case.

I've changed it to make use of the calling Parser instance, and it
now saves the links properly where one might expect.