Context: Protostellar jets in high-mass young stellar objects (HMYSOs) play a key role in the understanding of star formation and provide us with an excellent 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 Telescope instruments SINFONI, CRIRES, and ISAAC. Information about the source and its immediate environment is retrieved with SINFONI. The technique of spectro-astrometry is performed with CRIRES to study the jet on au scales. The parsec-scale jet and its kinematic and dynamic properties are investigated using ISAAC. Results: The SINFONI spectra in the $H$ and $K$ band are rich in emission lines 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-scale photocentre displacements ($11-15$,au). This allows us to constrain the kinematics of the au-scale jet and to derive its position angle ($sim216^{circ}$). ISAAC spectroscopy reveals H$_2$ emission along the parsec-scale jet, which allows us to infer kinematic and dynamic properties of the 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 roughly constant for the formation history of the young star. A tentative estimate of the ionisation fraction is derived for the massive jet by comparing the radio and 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 the radio jet only delineates a small portion of it.
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