Hi... my 91 accord with automatic tranny has a new starter and a new lock cylinder, and the starter signal fuse is good. nevertheless the starter is not engaging when I turn the key. Any ideas what the problem might be?

 

First, check whether the small wire at the starter actually gets live +12v when you turn the key. Unless the new starter is bad right-out-of-the-box, I bet you don't have a start signal on that wire. Then you start looking at components in-between the key and the starter.

Neutral-safety switch? Do the PRNDL lamps light up correctly?

How about the lock cylinder? That's just the mechanical key & cylinder, right? How do you know the electrical part of the switch is OK?