Well, actually, yes, that is how you play it. But there's set routines to the songs. Para Para is a solo club dance style. The game is based on that dance style. So, in the home version there's videos of the routines you can follow. In the second version of the arcade game, there's an animated character you can follow.
They didn't really bother to include the dancers in the first versions of the game because if you did para para at all, they'd be dances that you know.
That game is ooooold though. Most people who do para para don't play it anymore except for nostalgia's sake. There's thousands of para para routines set to eurobeat, trance music, and hyper techno. There's DVD series for it (Para Para Paradise, Gazen Para Para, Happy Paradise, Chozen Para Para, Para Para Panic, We Love Techpara. . . it goes on and on) to learn the dances from. Obviously, people in Japan have the luxury of being taught the dances at clubs.
The dance style is done in unison, except for some elements of freestyle in a few songs. Para Para was HUGE in the late 90's and early 2000s, but now just enjoys a modest popularity. It comes into the spotlight every couple of years and then dies off again.
The dance style is pretty heavily tied into the ganguro culture of the late 90's/early 2000s and its cousin gyaru culture.
In Missouri, where I'm moving from, I ran a para para performance team and we went to cons and hosted panels and did some performances of popular songs. It's a LOT of fun. And there's definitely more to it than the video game.