Technically speaking, the Enterprise only kept it's original registry number through special order from starfleet operations due to it's highly distinguished service record of the vessel and it's crew.
There have been multiple ships to bear the same name over the course of several generations. To name a few examples: Saratoga, Lexington, Yorktown, Excalibur, Constellation
The Saratoga from ST: IV was a different ship and had a different registry from Sisko's Saratoga in Emissary.
When it comes to naming a new starship, the fact that there have been other ships to bear the name is not what's special. It takes a special dispensation from starfleet to keep an original registry number.
It still perplexes me how, in universe at least, the Enterprise earned the distinction of being the only Federation starship to have suffixes added to its registry, in addition to becoming "the flagship of the fleet." Other starships undoubtedly had equally distinguishing careers as the Enterprise; after all, you don't name a ship after one with a bad recond (usually). What was it that made the Enterprise "special?" Real world reason was, of course, "Beacuse Gene Roddenberry said so."
As for your photoshop, Parallax, I'd go for the NCC-75633.