  We derive a new constraint on the expansion history of the Universe by
applying the cosmic chronometers method, studying the age evolution of
high-redshift galaxies with a full-spectral-fitting approach. We select a
sample of 39 massive ($log(M/M_\odot)>10.8$) and passive
($log(sSFR/yr^{-1})<-11$) galaxies from the data release 4 of the VANDELS
survey at $1<z<1.5$, combining different selection criteria to minimize the
potential contamination by star-forming outliers. We perform
full-spectral-fitting jointly on spectra and photometry of our sources with the
code BAGPIPES, without any cosmological assumption on the age of the
population. The derived physical properties of the selected galaxies are
characteristic of a passive population, with short star formation timescales
($<\tau>=0.28\pm0.02$ Gyr), low dust extinction ($<A_{V,dust}>=0.43\pm0.02$
mag), and sub-solar metallicities ($<Z/Z_{\odot}>=0.44\pm0.01$). The ages show
a decreasing trend with redshift compatible with a standard cosmological model,
even if no cosmological constraint is assumed in the fit, and a clear
mass-downsizing pattern. Testing the impact of the star formation history on
the results, we find only a maximum 2\% fluctuation in age and metallicity. By
fitting the median age-redshift relation with a flat $\Lambda$CDM model and
assuming a Gaussian prior on $\Omega_{M,0}= 0.3\pm0.02$ from late-Universe
probes, we obtain $H_0=67_{-15}^{+14}\:km\:s^{-1}\:Mpc^{-1}$. In the end, we
derive a new estimate of the Hubble parameter with the cosmic chronometers
method, $H(z=1.26)=135\pm65\:km\:s^{-1}\:Mpc^{-1}$ including statistical and
systematic errors. While the error budget is currently dominated by the
scarcity of the sample, this work proves the potential strength of the cosmic
chronometers approach up to $z>1$, especially in view of incoming large
spectroscopic surveys like Euclid. (abridged)
