Gravity can no longer be considered to be pulling downwards in a parallel fashion all along the flightpath. The parabolic model does not hold, and we have to change to an elliptical one. If we look at a parabola, we find the two ends of the curve get more and more parallel, but never actually make it. Thus our flight path can become an orbit. Think about what happens.
Gravity pulls it down all the time, and of course it does move down.
However its rate of fall matches the fall in ground level due to the Earth's curvature. It has made a circular orbit of the Earth. If the velocity or the launch angle were not absolutely spot on the orbit would have been elliptical, depending on how much in error the launch conditions were. You may think this is a silly analogy, but in 1997 a replica of the first satellite, Sputnik 1 was released by the crew of the Mir Space Station, to celebrate the 40th anniversary of the launch. Cosmonaut Pavel Vinogradov launched Sputnik 40, by hand.