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.
Something occured to me when I logged off last night about the Sao Paulo/Defiant-A, and I'd have to watch "The Dogs of War" and "What You Leave Behind" again to be sure. However, I don't recall the crew ever referring to the ship as as either the Sao Paulo or Defiant in dialogue after her introduction, despite the fact the Sisko was given permission to rename the ship. As I said, I'll have to watch the last two eps of DS9 to confirm my suspicions.