In this work, an extensive Metropolis Monte Carlo simulation is performed to investigate the steady-state magnetic and thermodynamic behaviour of a trilayered spin-1/2 Ising ferrimagnet with square monolayers, driven by external Gaussian random magnetic field with certain spatio-temporal variations. Such thin magnetic systems are interesting subjects for simulational studies as they exhibit compensation phenomenon. Here, two distinct theoretical atoms, A and B make up the ABA and AAB type of configurations. In ABA, A-atoms make up the surface layers and the mid-layer is composed up of B atoms. While, in AAB, A-atoms make up the top and midlayer while the bottom layer is made up of B-atoms. The like atoms (A-A and B-B) ferromagnetically interact and the unlike atoms (A-B) interact antiferromagnetically. For the time-dependent external Gaussian random field, the mean is zero always and the standard deviation is increased to unity in steps. Depending upon the strength of the external Gaussian random field, the compensation and critical points shift and steady-state magnetic behaviours shift between different distinct type of ferrimagnetic behaviours. The compensation phenomenon even vanishes after crossing a finite threshold of standard deviation of the magnetic field for particular choices of the other controlling parameters. Like Chandra S. [Phys. Rev. E 104, 064126 (2021)], in the Hamiltonian parameter space of both the configurations islands of ferrimagnetic phase without compensation appear within the phase area with compensation of field-free case. The areas of such islands grow with increasing standard deviation of the external field, σ, obeying the scaling relation: f (σ, A(σ)) = σ −b A(σ) with bABA = 1.913 ± 0.137 and bAAB = 1.625 ± 0.066 .