Finding the equation of a curve that you want your object to move along is one way to accomplish what you want, but probably not the best.
Instead, one usually keeps track of local properties of an object (velocity, acceleration) and then uses these values to update the object's position every frame.
Since you mentioned a parabola I am assuming that you are throwing a ball in 2D and you want it to fall down along the y-axis. So, your object has constant acceleration in the y-direction (let's call that
g) and no acceleration in the x-direction. When the object is thrown it is given some velocity, let's call that
Then, every frame of your application you would add the object's acceleration to its velocity, and then add it's velocity to it's position. Something like:
vy += g;
x += vx;
y += vy;
Do this every frame and your ball will start to move. There is a lot more to know about this, but it's a start.