Starting from the Gaia DR3 HR diagram, we derive the star formation history (SFH) as a function of distance from the Galactic plane within a cylinder centred on the Sun with a 200 pc radius and spanning 1.3 kpc above and below the Galaxy's midplane. We quantify both the concentration of the more recent star formation in the Galactic plane, and the age-related increase in the scale height of the Galactic disc stellar component, which is well-described by power laws with indices ranging from 1/2 to 2/3. The vertically-integrated star formation rate falls from $(1.147 \pm 0.039)\times 10^{-8}\, \text{M}_\odot \, \text{yr}^{-1} \, \text{pc}^{-2}$ at earlier times down to $(6.2 \pm 3.0) \times 10^{-9}\, \text{M}_\odot \, \text{yr}^{-1} \, \text{pc}^{-2}$ at present times, but we find a significant peak of star formation in the 2-3 Gyr age bin. The total mass of stars formed per unit area over time is $118.7 \pm 6.2\, \text{M}_\odot \, \text{pc}^{-2}$, which is nearly twice the present stellar mass derived from kinematics within 1 kpc from the Galactic plane, implying a high degree of matter recycling in successive generations of stars. The method is then modified by adopting an age-dependent correlation between the SFH across the different slices, which results in less noisy and more symmetrical results without significantly changing the previously mentioned quantities. This appears to be a promising way to improve SFH recovery in external galaxies.