Jets around low- and intermediate-mass young stellar objects (YSOs) contain a fossil record of the recent accretion and outflow activity of their parent star-forming systems. We aim to understand whether the accretion/ejection process is similar across the entire stellar mass range of the parent YSOs. To this end we have obtained optical to near-infrared spectra of HH 1042 and HH 1043, two newly discovered jets in the massive star-forming region RCW 36, using X-shooter on the ESO Very Large Telescope. HH 1042 is associated with the intermediate-mass YSO 08576nr292. Over 90 emission lines are detected in the spectra of both targets. High-velocity (up to 220 km s-1) blue- and redshifted emission from a bipolar flow is observed in typical shock tracers. Low-velocity emission from the background cloud is detected in nebular tracers, including lines from high ionization species. We applied combined optical and infrared spectral diagnostic tools in order to derive the physical conditions (density, temperature, and ionization) in the jets. The measured mass outflow rates are Ṁjet ~ 10-7M⊙ yr-1. It is not possible to determine a reliable estimate for the accretion rate of the driving source of HH 1043 using optical tracers. We measure a high accretion rate for the driving source of HH 1042 (Ṁacc ~ 10-6M⊙ yr-1). For this system the ratio Ṁjet/Ṁacc ~ 0.1, which is comparable to low-mass sources and consistent with models for magneto-centrifugal jet launching. The knotted structure and velocity spread in both jets are interpreted as fossil signatures of a variable outflow rate. While the mean velocities in both lobes of the jets are comparable, the variations in mass outflow rate and velocity in the two lobes are not symmetric. This asymmetry suggests that the launching mechanism on either side of the accretion disk is not synchronized. For the HH 1042 jet, we have constructed an interpretative physical model with a stochastic or periodic outflow rate and a description of a ballistic flow as its constituents. We have simulated the flow and the resulting emission in position–velocity space, which is then compared to the observed kinematic structure. The knotted structure and velocity spread can be reproduced qualitatively with the model. The results of the simulation indicate that the outflow velocity varies on timescales on the order of 100 yr.