Ooh! You want a better trajectory? is that it?

You can achieve "some sort" of parabolic movement simply by making the velocity vector of the particle always pointing at the player. To give it a little more movement, make the velocity vector's amplitude increase gradually, this way the particle will "fall" a little (to the bottom of the screen) before gaining velocity towards the player. (This is known as the predator-prey model)

Another possibility is to make the player something like a sun and the particle a tiny planet. Of course, you're not using the physics system, so you have to code this interaction yourself. But it's not difficult! Basically, you have to make tiny incremente to the particle's velocity towards the player per time unit (it's slightly different from the first approach). To make this solution more like a sun-planet system, make the increment increase with the distance to the player.

I don't know squat about coding particles, but at least I know a little about physics yay


edit: ...make the increment decrease with the distance... Because we want the particle to get more accelerated as it gets closer to the player.