How did pilots maintain the Vomit Comet's parabolic trajectory?
The NASA Vomit Comet was flown manually.
From The dynamics of parabolic flight: flight characteristics and passenger percepts

The control inputs required by the pilots of a parabolic flight aircraft are relatively simple, although precision is required to make the 0 g phase as close to freefall as possible, and care is required not to exceed the load limits of the aircraft which at times flies near its maximum rated speed. The pilots modulate lift L with the elevators and wings, which indirectly changes attitude θ, and thrust T with the engines (Figure 5).



....The pitch angle of the aircraft is not directly controlled by the pilots, but is a secondary result of other control inputs. During level flight the wings produce an upward lift, while the tail-mounted elevators produce a small downward component of lift in order to maintain a desired pitch angle. When flight conditions change, such as when the pilots deflect the elevators down during freefall, the relative forces from these two surfaces change. Since their moment arms are different, this causes a shift in the center of lift so that it no longer coincides with the aircraft center of gravity, resulting in a torque that causes a pitch rotation of the aircraft.


..At approximately 225 KT IAS (360 KT TAS, 185 m/s, Mach 0.61), when the aircraft is pitched nose-up 45°, the pilots commence the 0 g parabola. They push forward on the control yoke (&quot;push over&quot;) to lower the angle of attack of the wings, which reduces wing lift, and simultaneously reduce power to a level just sufficient to overcome drag.


...After 25 seconds, at the end of the parabola, when the nose is pitched down 45° and airspeed is close to 350 KT IAS, the pilots pull up (pull back on the control yoke) and increase thrust to change the aircraft’s downward velocity into upward velocity, and restart the cycle.