Yet-----------that is just what he says!
NO!...That is what Hosty says he said.... At the time of the interrogation (3:15 pm 11/22/63) Lee probably wasn't too concerned with being precise.... He was under the impression that he was a suspect in the death of some cop. I doubt that he was very careful in making sure he was precise in recalling his actions at the TSBD...because I'm sure he knew that there was no cop killed there.
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!