[Dead Parrot]

My first thought on reading this was worn/bad brushes in the starter. I did read where it is a new/rebuilt starter but when I replaced the brushes in mine, it acted similar for the first several starts until the new brushes settled in.

There are a set of contacts in the solenoid that switch in battery power to the starter. If those are shot, could also cause your problems. Once the solenoid clicks, all of the other safety switches are out of the picture as they all have to work before the solenoid will click.

Also 2nd the thought to recheck the battery cables and connections. A new good quality set of cables are good insurance. If the old cables turn out good, hang them on the wall for your next project.