The True Nature of Gravity

Thus, if this is true, then objects move in parabolic motion, because the integral of a constant acceleration gives a linear velocity, for which the integral gives a parabolic position.

Looking at Newtonian Gravitation, if you want to integrate with respect to time t, then you are stating that the Force F is some function where F(t) = GMm/r(t)^2.
Given that F(t) is a force, it could also be appropriately written as ma(t), where acceleration is a function due to time, which is appropriate in gravitation. Thusly, after dividing out the constant mass, and integrating with respect to time, we have
v(t) = integral of GM/r(t)^2 dt
z(t) = integral of v(t) dt

Which shows that for two objects, the acceleration is a function with respect to the distance they are away from each other, assuming they are in a state where movement is possible, and not truly constant, and that velocity is non-linear.

Acceleration is considered constant due to the fact that the change in the distance, r, is so insignificantly small that it need not be considered in most practical uses of the equation.