  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.
