I believe not, as all they are asking for is to go through your lot to access common property on the other side of your lot that is accessible from the other side, you may have exclusive use to the common property but it is outside your lot not inside and is accessible from the other side.
As an owner you are paying your share of the cost hence allowing them access would save you ….but no one can force you to let them have an easier access by going through your lot.
What are next? are they going to paint the outside of the building through your windows to save on scaffolding?
We all on this site have opinions do not trust anyone seek qualified advice if you do not want them walking through your lot.