Do I build three parties of NPCs all built with PC Race, Class and Background features
If they might ever be used by players as alternate PCs (like TPK backups), or by guest players, or perhaps as pregens at a convention or organized play even to or something, sure. In fact, you might have the players create 'alternate' or back-up characters that you then use as one of the rival parties - that could be fun.
or do I build them as monsters with a unique feature or two to capture the feel of a PC class without giving them full PC stats?
If they'll occasionally fight the PCs, yes.
If they're only going to be interacted with now & then, you can just stat out their social abilities - and, more importantly, give them names, compelling backstories/motivations/intra-party-tensions/etc. If they're going to be the targets of opposed checks in exploration ('race to the treasure' might occasionally call for some such contests), stat out those skills.
A "monster" write-up should be adequate for all three, if it's uncertain which sorts of scenes are likely.
If, OTOH, they're just going to be in the background, providing time pressure, you needn't stat them out, at all.
Or, another option might be to snag myself a copy of the Essentials Box and build these characters as Sidekicks?
Those rules sound pretty limited and focused on NPCs to be played along-side PCs, so probably not suitable.
Anyone have experience building rival adventuring parties? How did you stat them up, and how did you like the results? Would you do it again the same way, or would you do something differently?
My experience back in the day of statting up rivals as full PCs did not go well. They were too much work for too little return, and too complicated to use in actual play (and I'd often find myself going 'off the reservation,' anyway, and breaking PC create rules to get them 'right' - hey, Giants in the Earth did it all the time).
Thing is, playing a PC is a modest challenge, players can concentrate on their one PC and run it at something like it's full potential. The DM - though we DMs are a cut above, obviously - OTOH, has the whole world, in concept, the environment, and the monsters to keep a handle on, and can't go running whole parties of PCs to that same level (OTOH, the DM can make his monsters/NPCs/DMPCs cooperate better than the usual herd-of-cats PC party).
So, yeah, on balance, if there's going to be direct conflict, monster stat (or other condensed stat-blocks) are better from multiple perspectives.