The starter receives a constant 12 volts directly from the battery via the heavy gauge cable no matter if the key is in or out of the ignition switch. The starter's solenoid connection gets 12 volts via a lighter gauge wire connected to the starter relay inside the PDC only when the ignition switch is in the Start position. Are you getting 12 volts at the starter solenoid via the lighter gauge cable when the ignition switch is in the Start position. If both connections are getting a solid 12 volts when the ignition switch is in the Start position I'd say the new starter is bad.
If you don't get 12 volts at the starter's solenoid via the lighter gauge wire when the ignition switch is in the Start position I'd start looking at the ignition switch or, more likely, the ignition switch actuator pin behind the ignition switch as the source of the problem. A way to confirm that is to feel/hear the starter relay inside the PDC. If it's not clicking when the switch is in the Start position I'd suspect the ignition actuator pin has broken.