Parabellum is a contraction of two Latin words: Para = "for" and Bellum = "war". Thus, any ammunition designed for war is, by definition, parabellum. The British designed the .303 and we designed the 30-06, .308. .30 Carbine and .223 for war but they didn't get the parabellum tag.
So far as I know, only the Germans have used that designation for military cartridges and even that appears to be somewhat inconsistant. The 9mm Lugar is a parabellum only in that it was designed specifcally as a military round but so was the 7x57, etc.
I clearly remember my confusion of cartridge designations some 45 years ago. Cartridges specified by bore diameter, .300, and groove diameter, .308, and both shooting the same bullet. A .357 mag and .38 Special shooting the same bullet. A .44 mag and .444 Marlin shoots a .429. A 38/40 is actually a .40 cal. The .32 Special shoots a .311 bullet. The .218 Bee, .219 Zipper, 220 Swift, 22-250, 221 Fireball, 222 and 223 Rem, 224 Wea., .225 Win. all shoot .224 bullets at different speeds. Arrrghh!