The Party is a Faction
But only if the PCs make it one.
In other words, the Party is a Faction so long as all PCs merge their own activities & Subfactions underneath its banner, and clearly delineate the Party as an entity that actually exists in the world. It might be framed as a warband, a traveling caravan, or a nascent government — just so long as it can encompass the Factions the PCs also run.
Any PC may represent the Party as its Leader for any situation requiring a Leader’s action.