What type of orbit results if the initial velocity vector results in an eccentricity $e=1$?

An open orbit classified as a parabola

The spectrum of Keplerian orbits is entirely dependent on the eccentricity ($e$), which in turn is related to the initial conditions (velocity and position) of the orbiting body relative to the central mass under the influence of gravity. When the eccentricity is exactly 1 ($e=1$), the total mechanical energy of the system is precisely equal to the escape velocity. This mathematical condition defines an open trajectory known as a parabola. In such a trajectory, the orbiting object has just enough energy to escape the gravitational influence of the central body completely, meaning it will never return to its vicinity in an idealized system. Orbits with $e>1$ are hyperbolic and also open, possessing more energy than required for escape.