We present independent determinations of cosmological parameters using the distance estimator based on the established correlation between the Balmer line luminosity, L(H$beta$), and the velocity dispersion ($sigma$) for HII galaxies (HIIG). These results are based on new VLT-KMOS high spectral resolution observations of 41 high-z ($1.3 leq$ z $leq 2.6$) HIIG combined with published data for 45 high-z and 107 z $leq 0.15$ HIIG, while the cosmological analysis is based on the MultiNest MCMC procedure not considering systematic uncertainties. Using only HIIG to constrain the matter density parameter ($Omega_m$), we find $Omega_m = 0.244^{+0.040}_{-0.049}$ (stat), an improvement over our best previous cosmological parameter constraints, as indicated by a 37% increase of the FoM. The marginalised best-fit parameter values for the plane ${Omega_m; w_0}$ = ${0.249^{+0.11}_{-0.065}; -1.18^{+0.45}_{-0.41}}$ (stat) show an improvement of the cosmological parameters constraints by 40%. Combining the HIIG Hubble diagram, the cosmic microwave background (CMB) and the baryon acoustic oscillation (BAO) probes yields $Omega_m=0.298 pm 0.012$ and $w_0=-1.005 pm 0.051$, which are certainly compatible -- although less constraining -- than the solution based on the joint analysis of SNIa/CMB/BAO. An attempt to constrain the evolution of the dark energy with time (CPL model), using a joint analysis of the HIIG, CMB and BAO measurements, shows a degenerate 1$sigma$ contour of the parameters in the ${w_0,w_a}$ plane.
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