Check the picture below.
so if we look at that picture, is a parabolic path of an object that lasted from x = 0 to till it hit the ground.
Now, when the time is 0, that is when x = 0, does the height of the object have to be 0?
well, nope, it could be, but it doesn't have to be, if the time is 0 and the height is 0, only happens when the object starts off the same level it ended up, an example of that will be a soccer ball, is on the ground, you kick it, goes parabolically and then it hits the ground again, at time 0, it was on the ground, so its height was indeed 0.
A case when it doesn't have to be a height of 0 is for example if you're in a building on the 3rd floor and your sister outside on the sidewalk forgot her soccer ball, so she asks you to get it for her and throw it from the window, so the ball started at a height of the third floor, that may be who knows 50 ft or 30 meters, so at time 0 the object wasn't at height 0.
BTW, if you notice on that picture below, the "initial height" is not 0 is actually about 125ft.