In some contexts, gravitational force leads to constant acceleration kinematics. In other contexts, the acceleration of an object may change.
In cases where the gravitational field changes very little over the distances involved, gravitational force can be treated as constant, so the kinematics that result are constant acceleration. In such a constant field environment, the field lines would not significantly diverge over a small distance if you were to draw them on paper. In other words, parallel field lines signify a uniform field.
Try to picture the earth from space, and you would see the gravitational field lines spreading out into space. But if you think of the field within the room you are in, the field lines would appear to be parallel lines from the ceiling to the floor. This approximation of the gravitational field as constant allows us to treat the weight of an object on the surface of the earth as a constant value, although weight slightly decreases the further one gets from the center of the Earth (we weigh a bit less on top of a mountain than we did in the valley).
In summary, the space at or near the surface of the Earth is an area in which gravitational force is for most practical purposes a constant force, i.e. constant acceleration, environment. For problem solving, remember that the model for motion within this environment is projectile motion or free-fall. If differences in position under consideration are large compared to the distance between the bodies, then the force will be noticeably diminished with increasing separation.
Gravity in a room: the curvature of the Earth is negligible at this scale, and the force lines can be approximated as being parallel and pointing straight down to the center of the Earth.