They didn't have the key.

The team turned to Parliament's locksmith for help and, with some difficulty, he was able to open the wood panel door, to reveal a tiny, stone-floored chamber, with a bricked-up doorway on the far wall.

Looks like the locksmith picked it maybe?

They didn’t, they made a new key. An old lock for a service door is likely to be trivial to crack these days.

Edit: from the Guardian: “Liz Hallam Smith, the team’s historical consultant from the University of York, said: “[...] Once a key was made for it, the panelling opened up like a door into this secret entrance.”

The locksmith got the door open. It doesn't say he had the key.

Nope, but the video doesn't mention how they acquired the key either, they just demonstrate how it opens. I'll be honest and say that I didn't read very far into the article either and just skimmed it while listening to the video.