Parliamentary privilege, as traditionally viewed in the UK constitution, grants MPs freedom of speech on what they say within the House of Commons. It doesn’t:
a) give them the right to run spies in the civil service; or
b) cover what they say or do outside of the House of Commons.
Its relevance to the Damian Green case, therefore, is rather limited.
Update Dec 3: Sam Coates at the Times has been doing some digging, and has found that – of course – my interpretation is correct:
Parliamentary privilege is a narrow beast. Article IX of the 1689 Bill of Rights guarantees that “Parliamentary proceedings” – anything said on the floor of the Chamber or published in Hansard – cannot be used in evidence against MPs during a prosecution. But, citing a 1999 committee report, it says Parliamentary privilege “does not embrace and protect the activities of individuals, whether members or non-members, simply because they take place within the precincts of Parliament.”
The report cites the precedent of Lord Cochrane, who was arrested in 1815 while sitting on the Government front bench in the Chamber, having escaped from prison. The arrest took place before the sitting of the of the House, and the Committee of Privileges concluded that no breach of privilege had taken place.