While Garyx started the war, it gradually expanded to involve all of dragonkind.
The Thousand-Year War is an event specific to the history of the Forgotten Realms setting.
According to The Grand History of the Realms p.8, the Thousand-Year War took place from -26,000 DR to -25,000 DR when an avatar of Garyx (according to the 3e Draconomicon, a chaotic evil dragon deity of fire and destruction) led red dragons against the giants of Ostoria. Over the thousand-year course of the war, it gradually expanded to involve "all of dragonkind".
More of the history of the war appears in FOR7 Giantcraft, which asserts that, according to the legends of the dwarves, the division of chromatic and metallic dragons had not yet occurred when the war ended. The dragons ended the war in a truce in order to prepare for the civil war which would lead to the division of chromatic and metallic dragons. In other words, it sounds like the dragons of this era had no problem fighting alongside each other.
Volo's Guide to Monsters p.19 suggests that dragons lived in relative peace with giants until Garyx led his followers to war. The resulting war broke this peace, which in turn created the scenario that even dragons who did not follow Garyx found themselves involved in a conflict which pitted all giants against all dragons. It sounds to me like a situation where all dragons found themselves with a common enemy, and the different types of dragon may have put their differences aside to form alliances with each other.
Mordenkainen's Tome of Foes p.11 suggests that Tiamat was also personally involved in the war, and slew one of the gods of the giants, who cursed her to be bound to Avernus. This led to the end of the war.
The Grand History of the Realms p.8 also asserts that a few thousand years earlier, around -29,500 DR, the Dragonfall War took place, sparked by conflict between followers of Bahamut and Tiamat that continues to this day. This suggests that the two were not always at war with one another, so it is not entirely unthinkable that chromatic and metallic dragons might fight alongside one another.
The overall sense that I get is that while evil dragon deities like Garyx and Tiamat were certainly involved in the war, they were not the sole leaders of all dragonkind, but all dragonkind at least fought on their side during the war. You had an entire generation of dragons who were born during the war and had never known a life where they were not at war with the giants.
It's unspecified whether Bahamut and the other dragon gods personally took part in the war. Since Garyx and Tiamat were directly involved in the war, it's entirely possible that more deities were also involved.