The rotation will not necessarily be parallel to the ground. The general motion will be a combination of rotation in a horizontal plane (the conical pendulum) and oscillation in a vertical plane (the simple pendulum).
If the support (pivot) is a fixed point, the motion you get depends on the starting conditions. If you launch the mass horizontally at the right speed, the motion will be a perfect horizontal circle. However, at any other speed the mass will oscillate vertically also.
If the support moves (eg oscillates up and down, or side to side, or in a horizontal circle), then it is possible to enhance the horizontal rotation while suppressing the vertical oscillation. (In fact, this is one way to get is started - a small up-and-down oscillation at the right frequency can turn into a large sideways swing of the bob.)
In either case, if the kinetic energy of horizontal rotational motion is very large, the vertical oscillation will be an insignificant wobble.