If you do make it to the Parliament buildings, keep an eye out for the various "grotesques" you'll see peeking out at you from all over the building. My favorite is pictured here. A worker explained to me that these are not gargoyles, because gargoyles always have their mouths open to direct water away from the building, but grotesques have closed mouths.
Apparently the Parliament cats are about 40 strays they started taking care of. I only saw one, sleeping on a chair, when I was there. You might have better luck.