Always hard to recommend 2player co-op games for a PS2 because I often play them on GC. But a couple come to mind:
The Warriors - Good CO-OP beatemup action
Guitar Hero - Although this would be an expensive kareoke option but I hear its fun in a group.
We <3 Katamari - I've always played it single player, but I hear it makes a good party game. This isn't so much a game of skill, but a game everyone can enjoy right off the bat.
What type of genre are you looking for? I can check which RPGs have multiplayer, there's an absolute ton of fighting games on the PS2 if that is what you want and they are always 2player (Guilty Gear, Marvel vs Capcom 2, Capcom vs SNK 2, etc).
I'd check if X-Men Legends 2 is co-op. I play it over the internet on my PC and it is a ton of fun. The internet code could use some work but the game is more fun than Diablo2 IMO and a similar experience to it. I'm assuming if it is as good as the PC version without the annoying internet hastle, it could be awesome.