The present work provides the results of the first six years of operation of the systematic night-sky monitoring at ESO-Paranal (Chile). The UBVRI night-sky brightness was estimated on about 10,000 VLT-FORS1 archival images, obtained on more than 650 separate nights, distributed over 6 years and covering the descent from maximum to minimum of sunspot cycle n.23. Additionally, a set of about 1,000 low resolution, optical night-sky spectra have been extracted and analyzed. The unprecedented database discussed in this paper has led to the detection of a clear seasonal variation of the broad band night sky brightness in the VRI passbands, similar to the well known semi-annual oscillation of the NaI D doublet. The spectroscopic data demonstrate that this seasonality is common to all spectral features, with the remarkable exception of the OH rotational-vibrational bands. A clear dependency on the solar activity is detected in all passbands and it is particularly pronounced in the U band, where the sky brightness decreased by about 0.6 mag arcsec-2 from maximum to minimum of solar cycle n.23. No correlation is found between solar activity and the intensity of the NaI D doublet and the OH bands. A strong correlation between the intensity of NI 5200A and [OI]6300,6364A is reported here for the first time. The paper addresses also the determination of the correlation timescales with solar activity and the possible connection with the flux of charged particles emitted by the Sun. Comment: 19 pages, 27 figures, accepted for publication in Astronomy & Astrophysics. Full resolution version at http://www.hq.eso.org/~fpatat/science/skybright/paperIII.pdf