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When you turn the key to the on position does the red light flash very fast? Like 5 times a second?
Does this vehicle have a remote start or alarm installed? Do you have a test light to test for power?
If you press the unlock on the remote to disable the starter inhibit will the vehicle start?
No to which part? The starter can be imobolized by the alarm or remote start. That is very common. It could be the ignition switch isn't working properly and not turning on the starter. If the theft light was flashing I would suspect a faulty PCM power relay. Those are what suspect to be the issue.
A new starter can be bad out of the box. If you had a test light we can test for power. If you want to to guess I would replace the ignition switch. If you want the diagnostic to be 100% you will need to have some tools for testing.
Then I would suggest replacing the ignition switch if you are 100% the pats isn't malfunctioning.