Conclusions. The CO (J= 3 -> 2) emission in IRAS 16342-3814 is interpreted in terms of a jet-driven molecular outflow expanding along an elongated region. The position-velocity diagram and the mass spectrum reveal a feature due to entrained material that is associated with the driving jet. This feature is not seen in other more evolved objects that exhibit more developed bipolar morphologies. It is likely that the jet in those objects has already disappeared since it is expected to last only for a couple hundred years. This strengthens the idea that water fountain nebulae are undergoing a very short transition during which they develop the collimated outflows that shape the circumstellar envelopes. The oscillating pattern seen in the CO high-velocity outflow is interpreted as due to precession with a relatively small opening angle. The precession period is compatible with the period of the corkscrew pattern seen at IR wavelengths. We propose that the high-velocity molecular outflow traces the underlying primary jet that produces such a pattern.