What exactly was your reason for replacing the starter to begin with?
I have seen cases where ignition key switches go rogue and cause the starter to remain energized after the key is let return to the run position.
Easiest way to check for that is to unplug the small wire from the solenoid and connect that wire to a test light. Then turn the key to START position. The test light should burn ONLY while the key is held in START mode. If the test light remains lit after you let the switch return to RUN mode........you have a defective key switch.

Let's start by posting model and spec # of the engine along with what part # was replaced. Count the teeth of both pinion gears and see if the match. I've seen wrong starters put on these several times.

Tested solenoid contact which was working as needed.

Got my spec data together and was removing starter from engine to count teeth. Seems the half-moon protector strap between attachment bolts was interfering with the operation of the engagement mechanism. Added washers, reinstalled and all works okay. Amazing what you find !!

Again, thanx to responees