Hold your finger on the starter relay and see if it clicks when somebody turns the key to start or use a voltmeter on the solenoid wire. If the relay clicks (it's subtle but you can barley feel it), the problem may be in the big cables or low battery voltage/amperage.
The starter needs a little spin for the solenoid and Bendix to seat all the way, if there isn't enough current to spin the motor, you may not even hear a solenoid click or chatter.
An easy test is to turn on your headlights and try a start, if they dim down to nothing it's likely your battery. the battery connections need to be clean and tight, use some solvent and sand paper or 3M pad to clean them up. Make sure the ground strap at the dipstick holder is clean and tight.
A new starter doesn't necessarily mean a good starter, especially a rebuild. The biggest problem with the starter is usually poor contact at the brushes, they are worn or covered in oil.
The wire from the starter relay to the solenoid can be a trouble spot, it is too small and runs down the side of the block and can cook.
Clean the oil off of the front of the starter (where it mounts to the bell housing) and the bell housing mount, this is your ground a coat of oil can cause grief.