The length of the first portion of the book serves to establish the great power and tradition of the Igbo culture.  Over hundreds and hundreds of years they had built a narrative of the world and its creation and how they fit into the larger perspective of the world.  Achebe uses this longer portion to serve as the introduction to this powerful and established culture.

The shorter second and third parts are shorter in part to demonstrate the incredible speed with which things did in fact fall apart.  From the introduction of the Anglican missionaries to the massive conflicts between white colonists and black tribespeople just didn't take very long.  The precipitous decline of this powerful and celebrated culture happened so quickly, so too do parts two and three of the book.

Why are parts two and three shorter than the first part in Things Fall Apart?

My feeling about it is that the first part is longer in order to introduce and celebrate Ibo culture.  Achebe had a pretty clear objective in writing the story, which was to point out the terrible loss of culture that came about because of the very pernicious effects of colonization of Africa at the hands of various European powers.

Once the stage has been set and the various and interesting aspects of the Ibo culture have been shown and established, the descent into chaos and the horrible loss of that traditional culture came very quickly, so too those portions of the book are shorter.