Context. The detection of quasi-periodic variability in active galactic nuclei in general, and in blazars in particular, is key to our understanding of the origin and nature of these objects as well as their cosmological evolution. PG 1553+113 is the first blazar showing an approximately two-year quasi-periodic pattern in its γ-ray light curve, which is also revealed at optical frequencies. Aims. Such quasi-periodicity might have a geometrical origin, possibly related to the precessing nature of the jet, or could be intrinsic to the source and related to pulsational accretion flow instabilities. In this work we investigate and characterise the high-resolution radio emission properties of PG 1553+113 on parsec scales in order to differentiate between these different physical scenarios. Methods. We monitored the source with the very long baseline array (VLBA) at 15, 24, and 43 GHz during an entire cycle of γ-ray activity in the period 2015–2017, with a cadence of about 2 months, both in total and polarised intensity. We constrained the jet position angle across the different observing epochs by investigating the total intensity ridge lines. Results. We find a core-dominated source with a limb-brightened jet structure extending for ∼1.5 mas in the northeast direction whose position angle varies in time in the range ∼40°−60°. No clear periodic pattern can be recognized in the VLBA light curves during 2015–2017 or in the 15 GHz Owens Valley Radio Observatory light curve during the period 2008–2018. The core region polarisation percentage varies in the range ∼1−4%, and the polarisation angle varies from being roughly parallel to roughly transverse to the jet axis. We estimate a rotation measure value in the core region of ∼−1.0 ± 0.4 × 10⁴ rad m⁻². The brightness temperature (T_B) is found to decrease as the frequency increases with an intrinsic value of ∼1.5 × 10¹⁰ K and the estimated Doppler factor is ∼1.4. Conclusions. Although the jet wobbling motion indicates that geometrical effects can produce an enhanced emission through the Doppler boosting modulation, additional mechanisms are required in order to account for the quasi-periodic variability patterns observed in γ rays. The intrinsic T_B value indicates that the total energy in the core region is dominated by the magnetic field.