Even if an object is moving in a parallel orbit to Earth's around the sun, if you "zoom in" to the point where it enters the Earth's sphere of influence, the influence of the sun is relatively small. It's roughly as if Earth and the object are free-falling together in empty space, with a small relative velocity. And Earth's escape velocity is the same velocity that would be reached by a stationary object "dropped" from infinitely far away. So relative to the Earth it will accelerate and hit the atmosphere with at least escape velocity.
(This is only an approximation, but it's a pretty good one because there's such a large difference between Earth's mass and the sun's.)
Another way to look at this is to simply observe that Newtonian physics is time-symmetric (neglecting effects like friction that dissipate energy). If you have a trajectory that arrives at the edge of Earth's atmosphere with a certain velocity, you can reverse it to get a trajectory that leaves with the opposite velocity (i.e. equal speed).
(This is only an approximation, but it's a pretty good one because there's such a large difference between Earth's mass and the sun's.)
Another way to look at this is to simply observe that Newtonian physics is time-symmetric (neglecting effects like friction that dissipate energy). If you have a trajectory that arrives at the edge of Earth's atmosphere with a certain velocity, you can reverse it to get a trajectory that leaves with the opposite velocity (i.e. equal speed).