In addition to no air resistance, the other assumption is that the range is small enough so you can ignore the curvature of the earth, and make a flat Earth approximation. That's what takes a trajectory that's really an ellipse and approximates it as a parabola. Then the horizontal component of velocity will be constant.
Thanks. I got a question on a test a while ago where I knew the velocity a ball was thrown horizontally off a roof and knew how far the ball went before it landed but I couldn't do the question because I was convinced horizontal velocity wasn't constant. Ah well now I know.