As far as i understand (from my British friends): They generally can compel people inside the UK to produce papers, but would not otherwise have jurisdiction to order much here.
Instead, they ordered a person in the UK to produce non-UK protected documents.
So what they did here is pretty extraordinary, and if everyone didn't hate facebook, would likely cause a very serious diplomatic incident.
One of the underlying problems is that they aren't six4three's documents to produce. They are not the owner or custodian of these documents.
They simply have access under a protected set of circumstances by court order.
This is quite literally compelled espionage: "Espionage or spying, is the act of obtaining secret or confidential information without the permission of the holder of the information."
This wasn't the UK government, though, but Parliament through its Commons Digital, Culture, Media and Sport Committee.
Parliament doesn't allow government ministers to be members of parliamentary committees, only backbenchers, as Committees are one of the main ways that Parliament holds the government to account.
In American usage, "the government" is the totality of the body which exercises sovereign authority. All MPs, minister or backbencher, are part of the government, because Parliament is a government body. Similarly, any official action by any part of Parliament is an action of the British government.
Bluntly, Parliament is Parliament. (Her Majesty's) Government overlaps it, but is not the same. The DCMS committee in particular scrutinises government, but isn't part of it. I guess you could describe Parliament as being able to exercise (a few) State powers, but they really aren't a branch of government on the US theory.
A committee is not of government as it will include MPs from all sides, and seeks non-partisan balance no matter which party is currently in government. The Digital, Culture, Media and Sport Committee currently has 5 members from each of the main parties and one from the LibDems.
Committees can, at times, be quite strident in their criticism of government.
Das wage ich zu bezweifeln.