Dual flush toilet woes
The push-rods between the buttons and the flush mechanism need to be adjusted because, as you've noticed, when you put everything together they're pushing on the flush mechanism and allowing water through - so they're a little too long.  

You'll probably find that the end of the rods witch connect to the buttons is threaded, and there's some form of locking nut to hold them in place.
Loosen the nuts a few turns, screw the rods in a few turns to shorten them, re-tighten the nuts, and try again ...