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 H ii galaxies (HIIG). These results are based on new VLT-KMOS high spectral resolution observations of 41 high-z (1.3 <= z <= 2.6) HIIG combined with published data for 45 high-z and 107 z <= 0.15 HIIG, while the cosmological analysis is based on the MultiNest Markov Chain Monte Carlo (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 per cent increase of the figure of merit. The marginalized best-fitting parameter values for the plane {{Omega}m; w0} = {0.249^+0.11^_-0.065_ ; -1.18^+0.45^_-0.41_} (stat) show an improvement of the cosmological parameters constraints by 40 per cent. Combining the HIIG Hubble diagram, the cosmic microwave background (CMB) and the baryon acoustic oscillation (BAO) probes yields {Omega}m = 0.298 +/- 0.012 and w0 = -1.005 +/- 0.051, which are certainly compatible - although less constraining - than the solution based on the joint analysis of Ia supernovae (SNIa), CMB and BAO measurements. 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 {w0, wa} plane.