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Context: Protostellar jets in high-mass young stellar objects (HMYSOs) play akey role in the understanding of star formation and provide us with anexcellent tool to study fundamental properties of HMYSOs.  Aims: We aim at studying the physical and kinematic properties of the near-IR(NIR) jet of IRAS\,13481-6124 from au to parsec scales.  Methods: Our study includes NIR data from the Very Large Telescopeinstruments SINFONI, CRIRES, and ISAAC. Information about the source and itsimmediate environment is retrieved with SINFONI. The technique ofspectro-astrometry is performed with CRIRES to study the jet on au scales. Theparsec-scale jet and its kinematic and dynamic properties are investigatedusing ISAAC.  Results: The SINFONI spectra in the $H$ and $K$ band are rich in emissionlines that are mainly associated with ejection and accretion processes.Spectro-astrometry is applied to the Br$\gamma$ line, and for the first time,to the Br$\alpha$ line, revealing their jet origin with milliarcsecond-scalephotocentre displacements ($11-15$\,au). This allows us to constrain thekinematics of the au-scale jet and to derive its position angle($\sim216^{\circ}$). ISAAC spectroscopy reveals H$_2$ emission along theparsec-scale jet, which allows us to infer kinematic and dynamic properties ofthe NIR parsec-scale jet. The mass-loss rate inferred for the NIR jet is$\dot{M}_\mathrm{ejec}\sim10^{-4}\mathrm{\,M_\odot\,yr^{-1}}$ and the thrust is$\dot{P}\sim10^{-2}\mathrm{\,M_\odot\,yr^{-1}\,km\,s^{-1}}$ , which is roughlyconstant for the formation history of the young star. A tentative estimate ofthe ionisation fraction is derived for the massive jet by comparing the radioand NIR mass-loss rates. An ionisation fraction $\lesssim8\%$ is obtained,which means that the bulk of the ejecta is traced by the NIR jet and that theradio jet only delineates a small portion of it.

Parsec-scale jets driven by high-mass young stellar objects. Connecting the au- and the parsec-scale jet in IRAS 13481-6124