We can be quite certain that Agent Hosty, given the magnitude of what was at stake, was not apt to be casual about the suspect's claimed whereabouts at the time of the assassination.
The reason he wrote "Then went outside to watch P. Parade" was that he had clearly heard the suspect tell Captain Fritz that he had gone outside to watch the Presidential parade.
And the reason this startling claim never made it into any official report was that it pointed to the rather unwelcome fact that the suspect had an alibi.
Not complicated!
Alan, I'm 100% certain that Lee Oswald never fired any gun from the TSBD that day.....But I think you are putting too much faith in Hosty's notes. Hosty was under a great deal of duress when he was ordered to get his butt over to the DPD and make sure Oswald never said anything to implicate Hoover or his EXTRA SPECIAL agents. Hosty is the only person who recorded that Lee went outside to watch the President's Parade....

But, I do welcome your honest debate ......