Why does diazo coupling preferentially occur at the para position?
Both steric and electronic factors play a role in this reaction.  Resonance structures such as those shown below favor electrophilic attack (the diazonium ion is an electrophile) at the ortho and para positions. Since there are twice as many ortho positions as para positions in N,N-dimethylaniline, on a purely statistical basis we might expect a 67:33 ortho/para product distribution if only electronic effects controlled the reaction.  Since diazonium coupling usually produces a majority of the para coupled product, steric factors must be playing a role as well.  The somewhat bulky dimethylamino group does make it more difficult for an electrophile to approach the ortho position and the diazonium ion is relatively large.  These two steric factors combine to reduce the amount of the ortho isomer and the para isomer predominates.