I just got round to reading the article. They say that both Labour and the Tories support the 'status quo' as far as the GRA is concerned:
It should also be noted that it was the Tories who originally proposed writing self-ID into law, although their manifesto now supports the status quo. Similarly, although Labour’s 2017 manifesto supported self-ID, as does Jeremy Corbyn personally, the party’s present manifesto also backs the status quo.
But this is what the Labour manifesto says about the GRA (on page 68):
Labour is committed to reforming the Gender Recognition Act 2004 to introduce self-declaration for transgender people
labour.org.uk/wp-content/uploads/2019/11/Real-Change-Labour-Manifesto-2019.pdf