OR if there is another way to get a tour/in to the building?
Yes, you can also book a variety of tour types that include the entry to and tour of Parliament. For an example, Google "grand city and parliament tour" and you'll find companies that offer a city tour plus the Parliament tour. Other tours pair the Parliament tour with a Danube cruise.
I took one such half-day bus tour of the city and really enjoyed it. As part of the booking, they provided our English language Parliament tour tickets. During the course of our city tour itinerary, our city guide brought us to Parliament, collected the entry tickets for the group, saw us into our timed tour where we continued with the Parliament guide, waited for our Parliament tour to end and continued with us on the rest of our city tour.
Those tour groups may be where many of the English tour slots are being sold to on the dates you are searching.