Comments from @aquirdturtle have led me to rewrite my answer and to realise that it was a question worth asking.
@ACuriousMind has likened the situation to a mass falling on the Earth. In that case the mass and Earth system loses gravitational potential energy and they both gain kinetic energy although almost all of it resides with the mass.
Carrying on with the gravitational analogy then perhaps it is an example of gravity assist or gravitational slingshot? That idea can be ruled out because the slingshot effect works because not only is the satellite moving but so is the planet.
This sort of set up described was used by Thomson to measure the specific charge $\frac e m$ of the electron where electrons were first accelerated and then deflected by an E-field (and a magnetic field).
So assume that an electron is travelling at constant velocity along a straight line which is parallel to the plates and that a long way away from the plates the electron has zero electric potential energy and some kinetic energy.
The electron is under the influence of the E-field due to the parallel plate arrangement and so experiences a force which accelerates it; thus increasing the kinetic energy of the electron whilst decreasing the potential energy of the electron.
Since the E-field due to a charged parallel plate arrangement is small outside the region of the parallel plates, this change in kinetic energy and the resulting change in the direction of the electron’s velocity are only significant when the electron is between the parallel plates.
As the electron leaves the parallel plate arrangement it will gain potential energy and lose kinetic energy. Eventually the potential energy of the electron will become zero.