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We provide a new $Y_{SZ} - M_{500}^{Y_X}$ scaling relation using a sample of clusters from the Planck Early Sunyaev-Zeldovich (ESZ) catalogue observed in X-rays by Chandra, and compare it to the results of the Planck collaboration from XMM-Newton observations of a subsample of the ESZ. We calibrate a mass bias for a subset of the Planck cosmological cluster sample using published weak-lensing data from the Canadian Cluster Cosmology Project and Multi Epoch Nearby Cluster Survey, for the new scaling relation as well as that from the Planck collaboration. We propose a novel method to account for selection effects and find a mass bias of $(1-b)=0.89\pm0.04$ for the new scaling relation, and $(1-b)=0.76\pm0.04$ for the Planck scaling relation. With these mass biases, we obtain $Y_{SZ} - M_{500}$ scaling relations that we apply to the full Planck cosmological cluster sample, to obtain new constraints on the cosmological parameters. We find fully consistent constraints regardless of the X-ray sample used, with $\sigma_8 = 0.77\pm0.02$, $\Omega_m = 0.31\pm0.02$ and $S_8 \equiv \sigma_8 \sqrt{\Omega_m / 0.3}=0.78\pm0.02$. We also provide constraints with a redshift evolution of the scaling relation fitted from the data instead of fixing it to the self-similar value. We find a redshift evolution significantly deviating from the self-similar value, leading to a higher value of $S_8=0.81\pm0.02$. We compare our results to those from recent analyses based on various cosmological probes, and find that our $S_8$ constraints are competitive with the tightest constraints from the literature. When assuming a self-similar redshift evolution, our constraints are in agreement with most late time probes and in tension with constraints from the CMB primary anisotropies. When fitting the redshift evolution from the data, we find no significant tension with results from either late time probes or the CMB., Comment: 16 pages, 17 figures, submitted to A&A