For understanding the diversity of jetted active galactic nuclei (AGN) and especially the puzzling wide range in their radio loudness, it is important to understand what role the magnetic fields play in setting the power of relativistic jets in AGN. We have performed VLBA phase-referencing observations of the radio-intermediate quasar IIIZw 2 to estimate jet magnetic flux by measuring the core-shift effect. Multi-frequency observations at 4 GHz, 8 GHz, 15 GHz, and 24 GHz were made using three nearby calibrators as reference sources. By combining the self-referencing core shift of each calibrator with the phase-referencing core shifts, we obtained an upper limit of 0.16 mas for the core shift between 4 and 24 GHz in IIIZw 2. By assuming equipartition between magnetic and particle energy densities and adopting the flux-freezing approximation, we further estimated the upper limit for both the magnetic field strength and poloidal magnetic flux threading the black hole. We find that the upper limit to the measured magnetic flux is smaller by at least a factor of five compared to the value predicted by the magnetically arrested disk (MAD) model. An alternative way to derive the jet magnetic field strength from the turnover of the synchrotron spectrum leads to an even smaller upper limit. Hence, the central engine of IIIZw 2 has not reached the MAD state, which could explain why it has failed to develop a powerful jet even though the source harbours a fast-spinning black hole. However, it generates an intermittent jet, which is possibly triggered by small-scale magnetic field fluctuations, as predicted by the magnetic flux paradigm. We propose here that combining black hole spin measurements with magnetic field measurements from the very-long-baseline-interferometry core-shift observations of AGN over a range of jet powers could provide a strong test for the dominant factor that sets the jet power relative to the available accretion power.